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ACCESSION NO. 



CLASSIFICATION 



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COMPARATIVE MAGNITUDES OF THE PLANETS- 

1. MERCURY. 5. URANUS. 

2. MARS. 6. NEPTUNE. 

3. VENUS 7. SATURN. 

4. EARTH 8. JUPITER. 



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■a*>*-W-L)jl OUr 



THE 



ELEMENTS 



THEORETICAL AND DESCRIPTIVE 



ASTRONOMY, 



gw the \w of (&tA\t%n mt\ ^m&tmw. 



BY 

CHARLES J. WHITE, A 

A8SI8TANT PROFESSOR OF MATHEMATICS IN HARVARDlCOiLEf E, 



SECOND EDITION, REVISED. 



PHILADELPHIA: 

CLAXTON, REMSEN & HAFFELFINGER 

1872. 







^ 



V* 



<\V 



Entered according to Act of Congress, in the year 18G9, by 

CHARLES J. WHITE, 

in the Clerk's Oilice of the District Court of the United States for the District of 

Maryland. 




PREFACE. 



I HAVE endeavored in this treatise to present the main 
facts and principles of Astronomy in a form adapted to the 
course of instruction in that science which is commonly 
given at colleges and the higher grades of academies. I 
have selected those topics which appear to me to be both 
the most important and the most interesting, and have 
arranged them in the order which experience has led me to 
believe to be the best. 

I have endeavored, in the descriptive parts of the work, 
to give the latest information upon every topic which is in- 
troduced. The distances and the dimensions of the heavenlv 
bodies, together with other numerical data which depend for 
their values upon the value of the solar parallax, are given 
to correspond with the new value which is now adopted in 
the American Ephemeris. The results of the observations 
made at the United States Naval Observatory upon the 
November showers of the last two years will be found in the 
Chapter on Meteoric Bodies. Some description has also 
been given of the spectroscope, and of the many interesting 
discoveries made with it concerning the constitution of the 
heavenly bodies, and, very recently, concerning the real 
motions of the stars. ' 

No clear conception of the processes by which most of the 
fundamental truths of Astronomy have been established can 
be attained without some knowledge of Mathematics. I 
have endeavored, however, to confine the theoretical discus- 
sions within the limits of that moderate mathematical know- 

3 



4 PREFACE. 

# 

ledge which may fairly be expected in those readers for 
whom the treatise has been prepared. Certain definitions 
and formulae, with which the student may possibly not be 
familiar, will be found in the Appendix; and, with the aid 
of these, I believe that any one who has had the slightest 
mathematical education can read every portion of the work 
without difficulty. 

The treatises which I have especially consulted in the 
preparation of this book are Professor William Chauvenet's 
admirable Manual of Spherical and Practical Astronomy, and 
an elaborate treatise on Descriptive Astronomy, by George F. 
Chambers, recently published in England. An interesting 
chronological history of Astronomy, taken from the latter 
work, will be found in the Appendix; and Plates II., III., and 
IV. are from the same source. Plate I. is from a French work 
by Camille Flammarion, entitled La Pluralite des Mondes 
Habites. I am indebted to my friend Professor George A. 
Osborne, of the Massachusetts Institute of Technology, for 
his kindness in revising my manuscript. 

I trust that this treatise, although prepared especially as a 
text-book for students, may be of interest to others who may 
wish to know the general principles and the present state of 
the science of Astronomy. 

C. J. w. 

Harvard College, 

Cambridge, Mass., 
March, 1872. 



CONTENTS. 



The Greek Alphabet Page 10 

CHAPTER I. 

GENERAL PHENOMENA OF THE HEAVENS. DEFINITIONS. THE 
CELESTIAL SPHERE. 

The heavenly bodies. Astronomy. Form of the earth. Diurnal mo- 
tions of the heavenly bodies. Right, parallel, and oblique spheres. 
Definitions. Theorems. The astronomical triangle. Spherical co- 
ordinates. Vanishing lines and circles. Spherical projections 11 

CHAPTER II. 

ASTRONOMICAL INSTRUMENTS. ERRORS. 

The clock : its error and rate. The chronograph. The transit instru- 
ment: its construction, adjustment, and use. The meridian circle. 
The reading microscope. Fixed points. The mural circle. The 
altitude and azimuth instrument. Method of equal altitudes. The 
equatorial. The sextant. The artificial horizon. The vernier. 
Other astronomical instruments. Classes of errors 28 



CHAPTER III. 

REFRACTION. PARALLAX. DIP OF THE HORIZON. 

General laws of refraction. Astronomical refraction. Geocentric and 
heliocentric parallax. The dip of the horizon 51 

CHAPTER IV. 

THE EARTH. ITS SIZE, FORM, AND ROTATION. 

Measurement of arcs of the meridian by triangulation. Spheroidal 
form of the earth. Its dimensions. Its volume, density, and 
weight. Rotation of the earth. Change of weight in different 
latitudes. Centrifugal force. The trade-winds. Foucault's pendu- 
lum experiment. Linear velocity of rotation 59 



CONTENTS. 

CHAPTEK V. 

LATITUDE AND LONGITUDE. 

Four methods of finding the latitude of a place. Latitude at sea. 
Reduction of the latitude. Longitude. Greenwich time by chro- 
nometers, celestial phenomena, and lunar distances. Difference of 
longitude by electric and star signals. Longitude at sea. Compa- 
■ rison of the local times of different meridians Page 72 

CHAPTER VI. 

THE SUN. THE EARTH'S ORBIT. THE SEASONS. TWILIGHT. THE 
ZODIACAL LIGHT. 

The ecliptic. Distance of the sun from the earth determined by transits 
of Venus. Magnitude of the sun. The earth's orbit about the sun. 
The seasons. Twilight. Rotation of the sun, and its constitution. 
The zodiacal light 83 

CHAPTER VIL 

SIDEREAL AND SOLAR TIME. EQUATION OF TIME. THE CALENDAR. 

The sidereal and the solar year. Relation of sidereal and solar time. 
The equation of the centre. The equation of time. Astronomical 
and civil time. The calendar 100 

CHAPTER VIII. 

UNIVERSAL GRAVITATION. PERTURBATIONS IN THE EARTH'S ORBIT. 

ABERRATION. 

The law of universal gravitation. The mass of the sun. The earth's 
motion at perihelion and aphelion. Kepler's laws. Precession. 
Nutation. Change in the obliquity of the ecliptic. Advance of the 
line of apsides. Diurnal and annual aberration. Velocity of light. 
Aberration a proof of the earth's revolution 107 

CHAPTER IX. 

THE MOON. 

The orbit of the moon, and perturbations in it. Variation of the 
moon's meridian zenith distance. Distance, size, and mass of the 
moon. Augmentation of the semi-diameter. The phases of the 
moon. Sidereal and synodical periods. Retardation of the moon. 
The harvest moon. Rotation ; librations and other perturbations. 
The lunar cycle. General description of the moon 120 



CONTENTS. 7 

CHAPTER X. 

LUNAR AND SOLAR ECLIPSES. OCCULTATIONS. 

Lunar eclipses. The earth's shadow. Lunar ecliptic limits. Solar 
eclipses. The moon's shadow. Solar ecliptic limits. Cycle and 
number of eclipses. Occultations. Longitude by solar eclipses and 
occultations Page 135 



CHAPTEE XL 

THE TIDES. 

Cause of the tides. Effect of the moon's change .in declination. General 
laws. Influence of the sun. Priming and lagging of tides. The 
establishment of a port. Cotidal lines. Height of tides. Tides in 
bays, rivers, &c. Four daily tides. Other phenomena 146 

CHAPTER XII. 

THE PLANETS AND THE PLANETOIDS. THE NEBULAR HYPOTHESIS. 

Apparent motions of the planets. Heliocentric parallax. Orbits of 
the planets. Inferior planets. Direct and retrograde motion. Sta- 
tionary points. Evening and morning stars. Elements of a planet's 
orbit. Heliocentric longitude of the node. Inclination of the orbit. 
Periodic time. Mercury. Venus. Transits of Venus. Superior 
planets: their periodic times and distances. Mars. The minor 
planets. Bode's law. Jupiter: its belts, satellites, and mass. Saturn 
and its rings. Disappearance of the rings. L T ranus. Neptune. 
The nebular hypothesis 155 

CHAPTER XIII. 

COMETS AND METEORIC BODIES. 

General description of comets. The tail. Elements of a comet's orbit. 
Number of comets and their orbits. Periodic times. Motion in 
their orbits. Mass and density. Periodic comets. Encke's comet. 
Winnecke's or Pons's comet. Brorsen's comet. Biela's comet. 
D' Arrest's comet. Faye's comet. Mechain's comet. Halley's 
comet. Remarkable comets of the present century. The great 
comet of 1811. The great comet of 1843. Donati's comet. The 
great comet of 1861. Meteoric bodies. Shooting stars. The No- 
vember showers. Height and orbits of the meteors. Detonating 
meteors. Aerolites. Connection of comets and meteoric bodies 187 



8 CONTENTS. 

CHAPTEK XIV. 

THE FIXED STARS. NEBULAE. MOTION OF THE SOLAJl SYSTEM. REAL 
MOTIONS OF THE STARS. 

Proper motions of the fixed stars. Magnitudes. Constellations. 
Constitution of the stars. Distance of the stars. Bessel's differen- 
tial observations. Real magnitudes of the stars. Variable and tem- 
porary stars. Double and binary stars. Colored stars. Clusters. 
Resolvable and irresolvable nebulae. Annular, elliptic, spiral, and 
planetary nebulae. Nebulous stars. Double nebulae. The Ma- 
gellanic clouds. Variation of brightness in nebulae. The milky way. 
Number of the stars. Motion of the solar system in space. Real 
motions of stars detected with the spectroscope Page 213 

APPENDIX. 

Mathematical definitions, theorems, and formulae 242 

Kirkwood's law 247 

Chronological history of astronomy 248 

Sketch of the history of navigation 255 

Table I. — Elements of the planets, the sun, and the moon 256 

" IE—The earth 257 

III.— The moon 257 

" IV.— Elements of the satellites 258 

" V.— The minor planets 259 

VI. — Schwabe's observations of the solar spots 261 

" VII. — Elements of the periodic comets 261 

" VIII. — Transits of the inferior planets 262 

" IX. — Stars whose parallax has been determined 262 

" X.— The constellations 263 

" XL — Examples of variable stars 266 

" XII. — Examples of binary stars 267 

Index 269 



THE GREEK ALPHABET. 

The following table of the small letters of this alphabet is 
given for the use of those readers who are unacquainted with 
the Greek language. 



a 


Alpha. 


V 


Nu. 


P 


Beta. 


i 


Xi. 


7 


Gamma. 





Omlcron. 


h 


Delta. 


n 


Pi. 


e 


Epsllon. 


P 


Rho. 


I 


Zeta. 


(7 


Sigma. 


Y[ 


Eta. 


T 


Tau. 


SovQ 


Theta. 


V 


Upsllon. 


i 


Iota. 


<£ 


Flii. 


X 


Kappa. 


X 


Chi. 


X 


Lambda. 


4 


Psi. 


P 


Mu. 


(d 


Omega. 



10 



ASTRONOMY. 



CHAPTER I. 

GENERAL PHENOMENA OF THE HEAVENS. DEFINITIONS. 

1. The heavenly bodies are the sun, the planets, the satellites 
of the planets, the comets, the meteors, and the fixed stars. 

The planets revolve about the sun in elliptical orbits, and the 
satellites revolve in similar orbits about the planets. The earth 
is a planet, as we shall see hereafter, and the moon is its satel- 
lite. The comets revolve about the sun in orbits which are 
either ellipses, parabolas, or hyperbolas. Comparatively little 
is known with any degree of certainty about the meteors; but 
it is probable that they too revolve about the sun. 

The sun, the planets, the satellites, and the comets constitute 
what is called the solar system. The fixed stars are bodies which 
lie outside of this system, and preserve almost precisely the 
same configuration from year to year. 

The heavenly bodies may be considered to be projected upon 
the concave surface of a sphere of indefinite radius, the eye of 
the observer being at the centre of the sphere. This sphere is 
called the celestial sphere. 

2. Astronomy is the science which treats of the heavenly 
bodies. It may be divided into Theoretical. Practical, and De- 
scriptive Astronomy. 

Theoretical Astronomy may be divided into Sjiherieal and 
Physical Astronomy. 

Spherical Astronomy treats of the heavenly bodies when con- 
sidered to be projected upon the surface of the celestial sphere. 
It embraces those problems which arise from the apparent diur- 

11 



12 FORM OF THE EARTH. 

nal motion of the heavenly bodies, and also those which arise 
from any changes in the apparent positions of these bodies upon 
the surface of the celestial sphere. 

Physical Astronomy treats of the causes of the motions of the 
heavenly bodies, and of the laws by which these motions are 
governed. 

Practical Astronomy treats of the construction, adjustment, 
and use of astronomical instruments. 

Descriptive Astronomy includes a general description of the 
heavenly bodies; of their magnitudes, distances, motions, and 
configuration ; of their appearance and structure ; of, in short, 
every thing relating to these bodies which comes from observa- 
tion or calculation. 

3. Form of the Earth. — We may assume, at the outset, that 
the form of the earth is very nearly that of a sphere. The fol- 
lowing are some of the reasons which may be given for such an 
assumption : — 

(1.) If we stand upon the sea-shore, and watch a ship which 
is receding from the land, we shall find that the topmasts remain 
in sight after the hull has disappeared. If the surface of the 
sea were merely an extended plane, this would not happen ; for 
the topmasts, being smaller in dimensions than the hull, would 

in that case disappear first. 
.^. The supposition that the sur- 

^^// s \ s ^^sff ^ ace °^ ^ e sea * s curve d> 

xg^v''''/..L \-"' ""^^^ however, fully accounts for 
H xT / I \ \ this phenomenon, as may be 

/ \ seen in Fig. 1. Let the curve 

Fig x CBG represent a portion of 

the earth's surface, and let A 
be the position of the observer's eye : it is at once evident that 
no portion of the ship, S, will be visible which is situated below 
the line AH, drawn from A tangent to the earth's surface at C. 
The same figure also shows why it is that when a ship is ap- 
proaching land any object on shore can be seen from the top- 
masts before it is seen from the deck. 

(2.) At sea, the visible horizon everywhere appears to be a 
circle. This also is easily explained on the supposition that the 



DIFRNAL MOTION. 



13 



earth is spherical in form ; for if, in Fig. 1, the line AH is 
turned about the point A, and is continually tangent to the 
sphere, the points of tangency, C, D, E. &c, will form the visible ^ 
horizon of the spectator, and will evidently constitute a circle. 

(3.) A lunar eclipse occurs when the earth is situated between 
the moon and the sun. Now the shadow which the earth at 
such a time casts upon the moon is invariably circular in form : 
and a body which in every position casts a circular shadow- 
must be a sphere. 

4. Diurnal motion of the heavenly bodies. — Two things will be 
noticed by an observer who watches the heavens during any 
clear night. The first is, that all the heavenly bodies, with the 
exception of the moon and the planets, retain constantly the 
same relative situation ; and the second is, that all these bodies, 
without any exception whatever, are continually changing their 
positions with reference to the horizon. Let us suppose the 
observer to be at some place in the Northern Hemisphere. A 
plane passed tangent to the earth's surface at his feet will be his 





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ps I \e 

1/ / / \b 
/ / /\ 




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\r /\ / 



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Fig. 2. 



sensible horizon, and a second plane, parallel to this, passed 
through the centre of the earth, will be his rational horizon. If 



14 DIURNAL MOTION. 

these two planes be indefinitely extended in every direction, they 
will intersect the surface of the celestial sphere in two circles; 
but the radius of the celestial sphere is so immense in compari- 
son with the radius of the earth, that these two circles will 
sensibly coincide, and will form one great circle of the sphere, 
called the celestial horizon. In other words, the earth, when 
compared with the celestial sphere, is to be regarded as only a 
point at its centre. 

Let Fig. 2, then, represent the celestial sphere, at the centre 
of which, 0, the observer is stationed. Let the circle HESW be 
his celestial horizon, of which If is the north point, S the south, 
E the east, and W the west. 

If he looks towards the southern point of the horizon, and 
watches the movements of some star which rises a little east 
of south, he will see that it rises above the horizon in a cir- 
cular path for a little distance, attains its greatest elevation 
above the horizon when it bears directly south, and then 
descends and passes below the horizon a little west of south. 
In the figure, abc represents the path of such a star. If he 
notices a star which rises more to the eastward, as at d, for 
instance, he will see that it also passes from the east quarter of 
the horizon to the west in a circular path, attaining, however, a 
greater altitude above the horizon than that which the star a 
attained, and remaining above the horizon a longer time. A 
star which rises in the east point will set in the west point, and 
will remain twelve hours above the horizon. Turning his atten- 
tion to some star which rises between the north and the east, as 
the star g, for instance, he will find that its movements are simi- 
lar to the movements of the stars already noticed, and that it 
will remain above the horizon for more than twelve hours. 
Finally, if he turns towards the north, he will see certain stars, 
called circumpolar stars, which never pass below the horizon, but 
continually revolve about a fixed point in the heavens, very near 
to which point is a bright star called the Pole-star, which, to the 
naked eye, appears to be stationary, though observation shows 
that it also revolves about this same fixed point. In the figure, 
Im represents the orbit of a circumpolar star. 

If the same course of observations be repeated on the follow- 



DIURNAL MOTION. 15 

ing night, the observer will find that the situations of the stars 
with reference both to each other and to the horizon are the 
same that they were when he first began to examine them ; the 
motions which have already been described will be repeated, the 
circumpolar stars will still revolve about the same point in the 
heavens, and, in short, all the phenomena of which he took 
note will again be exhibited. 

5. Inferences. — Three important truths are proved by a series 
of observations similar to that which we suppose to have been 
made. 

(1.) The points at which the path of each star intersects the 
horizon remain unchanged from night to night, as long as the 
geographical position of the observer remains the same. 

(2.) All the stars, whether they move in great or in small 
circles, make a complete revolution in identically the same in- 
terval of time; — that is to say, in twenty-four hours. 

(3.) If we call that point about which the circumpolar stars 
appear to revolve the north pole of the heavens, and call the 
right line drawn from this point through the common centre of 
the earth and the celestial sphere the axis of the celestial sphere, 
the planes of the circles of all the stars are perpendicular to this 
axis. 

Whether, then, the earth remains at rest, and the celestial 
sphere rotates about its axis, as above defined, or the celestial 
sphere remains at rest, and the earth rotates within it on an axis 
of its own, one thing is certain : the axis of rotation preserves 
in either case a constant direction. If the celestial sphere 
rotates about its axis, this axis always passes through the same 
points of the earth's surface ; and if the earth rotates within the 
celestial sphere, its axis of rotation is constantly directed to the 
same points on the surface of the celestial sphere. 

We shall see hereafter the reasons which have led to the 
adoption of the theory that these apparent motions of the stars 
are really due to the rotation of the earth upon its own axis. 
A.t present, for the sake of convenience in description, we shall 
consider the earth to be at rest, and shall speak of the apparent 
motions of the heavenly bodies as though they were real. 

6. Farther observations. — Let us now suppose that the observer 



16 DIURNAL MOTION. 

leaves the place where he has hitherto been stationed, and travels 
in the direction of the point about which the circumpolar star? 
have appeared to revolve, and which we have called the north 
pole of the heavens. The general character of the phenomena 
which he observes will not be changed ; but he will notice a 
change in this respect : the elevation of the north pole above 
the horizon will continually increase as he travels towards it, 
and the planes of the circles of the stars, remaining constantly 
perpendicular to the axis of the celestial sphere, will become 
less and less inclined to the plane of the horizon. The conse- 
quence of this will be that stars which are near the southern 
point of the horizon will remain a shorter time above the hori- 
zon, and will finally cease to appear ; while in the northern 
quarter of the heavens the number of stars which never pass 
below the horizon will continually increase. Finally, if we sup- 
pose the observer to go on until the north pole is directly above 
his head, the stars which he sees will neither set nor rise, but 
will continually move about the sphere in circles whose planes 
are parallel to the plane of the horizon. In such a situation, it 
is evident that half of the celestial sphere will be perpetually 
invisible to him. Referring to Fig. 2, such a state of things is 
represented by supposing the line OP to be moved up into coin- 
cidence with OZ, the planes of the circles abc, def, &c, still re- 
maining perpendicular to OP. The stars a and d will lie con- 
tinually below the horizon, and the stars g and / continually 
above it. 

Such a sphere as this just now described, where the planes of 
revolution are parallel to the plane of the horizon, is called a 
parallel sphere. 

If the observer, instead of travelling towards the north pole, 
travels directly from it, its elevation above the horizon will con- 
tinually decrease, and the obliquity of the planes of the circles 
to the plane of the horizon will continually increase, until he 
will at length reach a point at which the north pole will lie in 
his horizon and the planes of the circles will be perpendicular to 
its plane. Eeferring again to Fig. 2, the line PO will in this 
case coincide with OH, and the arcs abc, def, &c, will have 
their planes perpendicular to the plane of the horizon. It is 



DEFINITIONS. 17 

evident that at this point every star in the celestial sphere will 
come above the horizon once in twenty-four hours, and that half 
of every circle will lie above the horizon, and half below it. 

The geographical position which the observer has now reached 
is some point on the earth's equator (Art. 7). Such a sphere as 
this, where the planes of the circles are perpendicular to the plane 
of the horizon, is called a right sphere. Besides the right and 
the parallel sphere, we have also the oblique sphere, where the 
planes of the circles are oblique to the plane of the horizon. 
Such a sphere is represented in Fig. 2. 

If the observer travels still farther in the same direction, the 
north pole will sink below his horizon, and the other extremity 
of the axis of the celestial sphere, called the south pole, will 
rise above it. There will be circumpolar stars revolving about 
this pole, and, in brief, all the phenomena which the observer 
noticed while travelling towards the north pole will be repeated 
as he travels towards the south pole. 

DEFINITIONS. 

7. We are now prepared to define certain points, angles, and 
circles on the earth and on the celestial sphere. 

The axis of the celestial sphere is, as we have already seen, all 
imaginary line drawn from the north pole of the heavens through 
the common centre of the earth and of the celestial sphere, 
and produced until it again meets the surface of the celestial 
sphere in the south pole. The points where this axis meets the 
surface of the earth are called the north pole and the south pole 
of the earth. 

That pole of the heavens which is above the horizon at any 
place is called the elevated pole at that place ; the other is called 
the depressed pole. 

The axis of the earth is that diameter which passes through 
the poles of the earth. 

The earth's equator is a great circle of the earth, whose plane 
is perpendicular to the axis. This circle is, of course, equidis- 
tant from the two poles, and divides the earth into two hemi- 
spheres. That hemisphere which contains the north pole is 

2 



18 DEFINITIONS. 

called the northern hemisphere, the other is called the southern 
hemisphere. 

Parallels of latitude are small circles of the earth whose planes 
are perpendicular to the axis. 

Terrestrial meridians are great circles of the earth passing 
through the poles. 

The latitude of any place on the earth's surface is its angular 
distance from the plane of the equator. This angle is measured 
by the arc of the meridian included between the place and the 
equator. Latitude is reckoned, either north or south, from 0° 
to 90°. 

The longitude of any place is the inclination of its own meri- 
dian to the meridian of some fixed station, and is measured by 
the arc of the equator included between these two meridians. 
Longitude is usually reckoned east or west of the fixed meri- 
dian, from 0° to 180°. The meridian of Greenwich, England, 
is most commonly taken for the fixed meridian, though the 
meridians of Washington, Paris, and other places are also taken 
for the same purpose. The fixed meridian is called the prime 
meridian. 

The arc of a parallel of latitude included between any two 
meridians is the departure between those meridians for that lati- 
tude. It is evident that if the departure between any two meri- 
dians is taken on two different parallels of latitude, the depart- 
ure at the greater latitude will be the smaller. 

Upward motion at any place is motion from the centre of the 
earth. Downward motion is towards the same point. 

The celestial horizon has already been defined (Art. 4). Lines 
drawn perpendicular to the plane of the horizon are called 
vertical lines. The vertical line at any place, if indefinitely 
prolonged, meets the celestial sphere in two points. The upper 
point is called the zenith, the lower point the nadir. 

The celestial meridian of any place is the great circle in which 
the plane of the terrestrial meridian of that place, when indefi- 
nitely produced, meets the surface of the celestial sphere. The 
axis of the sphere divides the celestial meridian into two semi- 
circumferences : that which lies on the same side of the axis as 
the zenith is called the upper branch of the meridian, and the 



DEFINITIONS. 19 

other is called the lower branch. The points where the celestial 
meridian and the celestial horizon intersect are called the north 
and the south point of the horizon, — the point which is the nearer 
to the north pole being the north point. The line in which 
the planes of these same two great circles intersect is called the 
meridian line. 

Vertical circles are great circles of the sphere which pas3 
through the zenith and nadir. That vertical circle the plane 
of which is perpendicular to the plane of the celestial meridian 
is called the prime vertical. The points in which the prime ver- 
tical cuts the horizon are called the east and the west point of the 
horizon ; and the line which joins these tw T o points is the east 
and west line. 

The celestial equator, also called the equinoctial, is the great 
circle of the sphere in which the plane of the earth's equator, 
indefinitely produced, meets the celestial sphere. 

The altitude of a heavenly body is its angular distance above 
the plane of the celestial horizon, measured on a vertical circle 
passing through that body. The zenith distance of the body is 
its angular distance from the zenith, and is evidently the com- 
plement of the altitude. 

The azimuth of a celestial body is the inclination of the verti- 
cal circle which passes through the body to the celestial meri- 
dian, and is measured by the arc of the celestial horizon in- 
cluded between this vertical circle and the celestial meridian. 
Azimuth may be reckoned from either the north or the south 
point of the horizon, and towards either the west or the east. 
Navigators usually reckon it from the north point in north lati- 
tude and the south point in south latitude, and east or west as 
the body is east or west of the meridian, thus restricting it 
numerically to values less than 180°. Astronomers, on the con- 
trary, usually reckon from the south point, to the right hand, 
from 0° to 360°. 

The amplitude of a heavenly body is its angular distance from 
the prime vertical w T hen in the horizon. It is reckoned from 
the east point when the body is rising, and from the w r est point 
when the body is setting, towards the north or the south as the 
body is to the north or the south of the prime vertical. 



20 DEFINITIONS. 

The vernal equinox is a certain fixed point upon the equinoc- 
tial. It is also called the first point of Aries. 

Hour circles, or circles of declination, are great circles of the 
sphere passing through the poles of the heavens. 

The right ascension of a heavenly body is the inclination of 
its hour circle to the hour circle which passes through the vernal 
equinox ; or it is the arc of the equinoctial intercepted between 
these two hour circles. Right ascension is usually reckoned in 
hours, minutes, and seconds (an hour being taken equal to 15° 
of arc), and is always reckoned to the eastward, from Oh. to 24h. 

The declination of a heavenly body is its angular distance 
from the plane of the celestial equator, measured on the circle 
of declination passing through the body. It is reckoned in 
degrees, minutes, &c, to the north and the south. The polar 
distance of a body is its angular distance from either pole, mea- 
sured on its hour circle. Usually, however, when we speak of 
the polar distance of a body, we mean its angular distance from 
the elevated pole. 

If, in Fig. 2, with the arc HP, which measures the altitude 
of the elevated pole, as a polar radius, we describe a circle about 
the pole as a centre, it is evident that the stars whose circles lie 
within this circle will never set. This circle is called the circle 
of perpetual apparition. It is equally evident that stars whose 
circles lie within a circle of the same magnitude, described about 
the depressed pole as a centre, will never come above the horizon. 
This circle is therefore called the circle of perpetual occidtation. 

The passage of a celestial body across the meridian is called 
its transit or culmination. When the body is within the circle 
of perpetual apparition, both transits occur above the horizon, 
one above the pole, the other below it. These are called the 
upper and the lower transit. For all bodies outside this circle, 
and not within the circle of perpetual occultation, the upper 
transit occurs above the horizon, the lower below it. For all 
bodies whatever, the upper transit occurs when the body crosses 
the upper branch of the meridian, and the lower transit when 
it crosses the lower branch. 

The hour angle of a heavenly body is the inclination of the 
circle of declination which passes through the body to the 



CELESTIAL SPHERE. 



21 



celestial meridian, and is measured by the arc of the celestial 
equator included between these two circles. Hour angles are 
reckoned positively towards the west, from the upper culmina- 
tion, from 0° to 360°, or Oh. to 24h. 

The hour angle of the sun is called solar time, and that of the 
first point of Aries sidereal time. The interval of time between 
two consecutive upper transits of the sun is called a solar day, 
and the interval between the upper transits of the first point of 
Aries is called a sidereal day. The celestial sphere apparently 
makes one revolution about the earth in a sidereal day. The 
solar day is, on the average, about 3m. 56s. longer than the 
sidereal day. 

8. Some of the preceding definitions are illustrated in the 
diagram, Fig. 3. In this figure is the position of the observer, 
HESW his celestial horizon, Pp the axis of the heavens, P 
the elevated and p the depressed pole, Zthe zenith, and JVthe 
nadir. The circle HZSN is the observer's celestial meridian. 
It may be noticed that this circle is at once a vertical and an 




hour circle. The circle EC WD is the equinoctial, and the circle 
EZWN, perpendicular to the meridian, is the prime vertical, 



22 THEOREMS. 

cutting the horizon in E and W, the east and the west point of 
the horizon. The equinoctial, being also perpendicular to the 
meridian, passes through the same points. If P is supposed to 
be the north pole, H is the north point of the horizon, and S the 
south point. 

Let A denote some celestial body. GA is its altitude, ZA its 
zenith distance, HG its azimuth as reckoned by navigators, and 
SG its azimuth as reckoned by astronomers, all these elements 
of position being determined by the arc of a vertical circle, ZG, 
passed through A. Let an arc of an hour circle, PB, be also 
passed through A. Then is AB its declination, PA its polar 
distance, and if V be taken to denote the position of the vernal 
equinox, VB is the right ascension of A. The right ascension 
may also be represented by the angle VPB, which the arc VB 
measures. The angle ZPB is the hour angle of A, and ZPV is 
the hour angle of the vernal equinox, or the sidereal time. 
This angle may also be designated as the right ascension of the 
meridian. 

The circle KH, drawn about P with the radius PH, is the 
circle of perpetual apparition. The star whose path is repre- 
sented by Im never passes below the horizon : I is its upper, m 
its lower culmination. SB represents the circle of perpetual 
occupation, and the stars whose paths lie, like or, within this 
circle, never come above the horizon. 

9. Theorem. — The sidereal time at any place is always equal to 
the siim of the right ascension and the hour angle of the same body. 
This is an important astronomical theorem, and is readily proved 
in Fig. 3, in which ZPV, the sidereal time, is the sum of ZPB, 
the hour angle, and VPB, the right ascension of the celestial 
body A. Any two of these three angles, then, being given, the 
third is readily obtained. 

Corollary. — When a celestial body is at its upper culmination 
at any place its right ascension is equal to the sidereal time at that 
place. This is evidently true, because the hour angle of a body 
when at its upper culmination is zero, and hence, from the theorem, 
the right ascension of the body is equal to the sidereal time. For 
example, in Fig. 3, if lis a body at its upper culmination, VPZ is 
both its right ascension and the right ascension of the meridian. 



THEOREMS. 




Fig. 4. 



10. Theorem. — The latitude of any place on the earth's surface 
is equal to the altitude of the elevated pole at that place. Let L 
(Fig. 4) be some place on the 
earth's surface, Pp the earth's 
axis, and EQ the equator. The 
line HE, tangent to the earth's 
surface at L, is the horizon, and 
Z the zenith, of L. According to 
the definition already given, LOQ 
is the latitude of L. Let the 
earth's axis be indefinitely pro- 
longed, and at L let the line LP' , 
parallel to the earth's axis, be also 
indefinitely prolonged. Owing 
to the immensity of the celestial sphere when compared with 
the earth, these two lines will sensibly meet at a common point 
on the surface of the celestial sphere, and this common point 
will be the elevated pole. The elevated pole, then, to an ob- 
server at L, will lie in the direction LP", and P'LH will be its 
altitude. 

Now we have, HLZ = POQ 

ZLP" = ZOP' 
.-.P"LH=LOQ 
which was to be proved. 

11. Theorem. — The latitude of a place is equal to the declination 
of the zenith of that place. This is easily seen in Fig. 4 : for the 
declination of any body or point is its angular distance from the 
plane of the celestial equator, and hence ZOQ in the declination 
of the zenith. 

Either of these theorems might be deduced from the other, 
but it is better to consider them as independent propositions. 

12. The Astronomical Triangle. — The spherical triangle PZA 
(Fig. 3) is called the astronomical triangle. It is formed by the 
arcs of the meridian of the place, and of the vertical circle and 
the hour circle passing through some heavenly body, which are 
included between the zenith of the observer, the elevated pole, 
and the position of the body as projected on the surface of the 
celestial sphere. The three sides are : ZP, the co-latitude of the 



24 DIURNAL CIRCLES. 

place; PA, the polar distance of the body; and ZA, its zenith 
distance. The three angles are : ZPA, the hour angle of the 
body; PZA, its azimuth; and PAZ, an angle which is rarely 
used, and which is commonly called the position angle of the 
body. 

The co-latitude and the zenith distance can evidently never 
be greater than 90°. The polar distance is equal to 90° minus 
the declination: and it is less than 90° when the body is on the 
same side of the celestial equator as the observer, but becomes 
numerically 90° plus the declination when the body is on the 
opposite side. In the former case the declination of the body 
is said to have the same name as the latitude, in the latter case, 
to have the opposite name. 

13. Diurnal Circles. — We have already seen that the apparent 
daily motions of the stars are performed in circles, the planes 
of which are perpendicular to the axis of the sphere. These 
circles are called diurnal circles. The phenomena which have 
been observed with reference to these circles (Arts. 4 and 6) are 
explained in Fig. 3. The circles of all the stars which rise in 
the arc of the horizon, ES, are evidently divided by the horizon 
into two unequal parts, the smaller of which in each case lies above 
the horizon. Hence these stars are above the horizon less than 
twelve hours. On the other hand, the greater portions of the 
circles of those stars which rise between E and H lie above the 
horizon, and the stars themselves are above it for more than 
twelve hours. 

Any body, then, whose declination is of the same name as the 
elevated pole will be above the horizon more than twelve hours, 
while a body whose declination is of a different name from that 
of the elevated pole will be above the horizon less than twelve 
hours. And further: those bodies which are in north declination, 
in other words, to the north of the celestial equator, will rise to 
the north of east and set to the north of west; while bodies in 
south declination will rise and set to the south of the east and 
the west point. 

When a star has no declination, that is to say, is on the celestial 
equator, it will rise due east and set due west, and will remain 
twelve hours above the horizon. 



SPHERICAL CO-ORDINATES. 25 

14. Right and Parallel Spheres. — When an observer travels 
towards the elevated pole, the radius of the circle of perpetual 
apparition, being equal to the altitude of the pole, continually 
increases, and the number of stars which never set increases in 
Lke manner. The number of stars which never rise also increases. 
Finally, if he reaches the pole, the celestial equator coincides 
with the horizon, the east and the west point disappear, and the 
bodies which are on the same side of the equator with the ob- 
server are perpetually above the horizon, and revolve in circles 
whose planes are parallel to it, while the bodies which are on 
the opposite side of the equator never rise. As he travels to- 
wards the equator, the circles of perpetual apparition and occul- 
tation alike diminish, the diurnal circles become more and more 
nearly vertical, and when he reaches the equator, the equinoctial 
becomes perpendicular to the horizon and coincides with the 
prime vertical, and the horizon bisects all the diurnal circles. 
At the equator, then, every celestial body comes above the hori- 
zon, and remains above it twelve hours. 

15. Spherical Co-ordinates. — The position of any point on the 
surface of a sphere is determined, as soon as its angular distances 
are given from any two great circles on that sphere whose po- 
sitions are known. Thus the geographical position of any point 
on the earth's surface is known, when we have determined its 
latitude and longitude; in other words, when we know 'its-an- 
gular distance from the equator and from the prime meridian; 
In like manner we know T the position of any point on the sur- 
face of the celestial sphere when either its altitude and azimuth, 
or its right ascension and declination, are given. In the first 
of these two systems of co-ordinates the fixed great circles are 
the celestial horizon and the celestial meridian, the origin of co- 
ordinates being either the north or the south point of the horizon. 
In the second system the fixed great circles are the equinoctial 
and the circle of declination which passes through the vernal 
equinox, and the origin of co-ordinates is the vernal equinox. 
In Fig. 3, if we know the arcs HG and GA, or the arcs VB 
and BA, we evidently know the position of A. 

16. Vanishing Points and Vanishing Circles. — Every one knows 
that as he increases the distance between himself and any object, 



26 VANISHING POINTS AND CIKCLES. 

the apparent magnitude of the object decreases ; and that, if he 
recedes far enough from it, it will be reduced in appearance to 
a point. Every one also knows that when he looks along the line 
of a railroad track the lines appear to converge, and that, if 
the track is straight, and the curvature of the earth does not 
limit his vision, the rails will ultimately appear to meet. 

These familiar illustrations will serve to show what is meant 
by a vanishing point. The actual distance between the rails of 
course remains the same; but the angle at the eye which this 
distance subtends decreases as the eye is directed along the 
track, until at last it ceases to subtend any appreciable angle at 
the eye, and the rails apparently meet. This point where the 
rails appear to meet is called the vanishing point of the two 
lines; and, in general, the vanishing point of any system of pa- 
rallel lines is the point at which they will appear to meet, when 
indefinitely prolonged. We have already seen (Art. 10) that 
the pole of the heavens is the vanishing point of lines drawn 
perpendicular to the equator, and the same may be said of the 
poles of any circle on the celestial sphere. For instance, the 
poles of the horizon at any place are the zenith and the nadir; 
and any system of lines perpendicular to the horizon will appa- 
rently meet, when prolonged indefinitely, in these two points. 
And again, the east and the west point of the horizon are the 
poles of the meridian, and lines drawn perpendicular to the 
meridian will have these points for their vanishing points. 

The same principle holds good w 7 hen applied to any system 
of parallel planes. They will appear to meet, when indefinitely 
extended, in one great circle of the sphere, and this circle is 
called the vanishing circle of that system of planes. The celes- 
tial horizon is, as has already been stated (Art. 4), the vanishing 
circle of the planes of the sensible and the rational horizon, and 
indeed of any number of planes passed parallel to them. The 
celestial equator is the vanishing circle of the planes of all the 
parallels of latitude, and, in short, every circle of the celestial 
sphere may be regarded as the vanishing circle of a system of 
planes passed perpendicular to the line which joins the poles of 
that circle. 

17. Spherical Projections ---The points and circles of either the 



SPHERICAL PROJECTIONS. Zr 

earth or the celestial sphere, or of both, may be projected upon 
the plane of any great circle of either sphere. The plane on 
which the projections are made is called the primitive plane, and 
the circle which bounds this plane is called the primitive circle. 
Several distinct methods of projection will be found in treatises 
on Descriptive Geometry. Of these, the most common are the 
orthographic, the stereo graphic, and Mercators projection. 

In the orthographic projection, the point of sight is taken in 
the axis of the primitive circle, and at an infinite distance from 
that circle. All circles whose planes are perpendicular to the 
primitive plane are projected into right lines; all circles whose 
planes are parallel to the primitive plane are projected into 
circles, each of which is equal to the circle of w T hich it is a pro- 
jection ; and all other circles are projected into ellipses. 

In the stereographic projection, the point of sight is at either 
pole of the primitive circle, and its distance from that circle is 
finite. In this projection every circle is projected as a circle, 
unless its plane passes through the point of sight, in which case 
it is projected into a right line. 

Mercator's projection is employed in the construction of charts 
representing the earth's projection. In this projection the 
parallels of latitude are represented by parallel right lines, and 
the meridians are also represented by parallel right lines, per- 
pendicular to the equator. The meridian projections are equi- 
distant, but the distance between the successive latitude projec- 
tions increases as w r e recede from the equator. The advantage 
offered by this projection to navigators is that the ship's track, 
as long as the course on which it sails is unaltered, is represented 
on the chart by a straight line, and that the angle which this 
line makes with each meridian is the course 



28 INSTRUMENTS, 



CHAPTER II. 

ASTRONOMICAL INSTRUMENTS. ERRORS. 

18. This chapter will be devoted to a general description of 
the common astronomical instruments, of the class of observa- 
tions to which each is adapted, and of the manner in which such 
observations are made. No attempt will be made to describe 
the elaborate mechanism by which, in many cases, the usefulness 
of the instrument is increased and its manipulation is facilitated ; 
but enough, it is hoped, will be said to enable the student to 
form a clear conception of the prominent features of each instru- 
ment which is described. There is no lack of excellent treatises 
on Astronomy, in which those who wish to investigate this subject 
more thoroughly will find all the details, w T hich the limits pre- 
scribed to this book will not permit to enter here, clearly and 
elaborately presented. 

THE ASTRONOMICAL CLOCK. 

19. The astronomical clock is a clock which is regulated 
to keep sidereal time, and is an indispensable companion to the 
other astronomical instruments. It is provided with a pendulum 
so constructed that change of temperature will not affect its 
length. The sidereal day at any place commences, as has al- 
ready been stated, when the vernal equinox is on the upper 
branch of the meridian of that place, and the theory of the 
sidereal clock is that it shows Oh. Om. Os. when the vernal equi- 
nox is so situated. Practically, however, it is found that every 
clock has a daily rate; that is to say, it gains or loses a certain 
amount of time daily. In order, then, that a clock may be re- 
gulated to sidereal time, it is necessary to know both its error 
and its daily rate; the error being the amount by which it is 
fast or slow at any given time, and the daily rate being the 



THE CLOCK. 29 

amount which it gains or loses daily; and knowing these, it 
is evidently in our power to obtain at any desired instant 
the true sidereal time from the time shown by the face of the 
clock. 

It is to be noticed further, that, except as a matter of con- 
venience, a small rate has no advantage over a large one; but 
it is very important that the rate, whether large or small, shall 
be constant from day to day; so that, of two clocks, one of which 
has a large ano> constant rate, and the other a small and varying 
one, the preference is to be given to the former. 

Clocks may be regulated to keep either local sidereal time or 
Greenwich sidereal time, or both. 

20. Error of the Clock. — To obtain the error of a clock on the 
local sidereal time at any observatory, we make use of the pro- 
position, already demonstrated (Art. 9), that the right ascension 
of any celestial body, when at its upper culmination on any 
meridian, is equal to the sidereal time at that meridian. The 
Nautical Almanac gives the right ascensions of more than a 
hundred stars which are suitable for observations for time. By 
means of an instrument, properly adjusted, we determine the 
instant when any one of these stars is on the meridian, and the 
time which the clock shows at that instant is noted. This is the 
clock time of transit, and the right ascension of the star, taken 
from the Almanac, is the true time of transit; and a comparison 
of these two times will evidently give us the amount by which 
the clock is fast or slow on local sidereal time. 

21. Daily Rate. — If, in a similar manner, we obtain the error 
of the clock on the next or on any subsequent day, the difference 
of these two errors will be the gain or loss of the clock in the 
interval; and hence, if we divide this difference by the number 
of the days and parts of days w r hich have intervened between 
the two observations, the quotient w r ill be the daily gain or 
loss. 

22. Chronograph. — The accuracy of astronomical observations 
i& much enhanced by recording the times of the observations by 
means of an electric current. A cylinder, about which a roll 
of paper is wound, is turned about on its axis with a uniform 
motion by the use of appropriate machinery. A pen is pressed 



30 



THE TRANSIT INSTRUMENT. 




Pig. 5. 



THE TRANSIT INSTRUMENT. 31 

down upon the paper, and is so connected with a battery thai, 
whenever the circuit is broken a mark of some kind is made upon 
the paper. The wires of this battery are connected with the 
sidereal clock in such a way that every oscillation of the pendu- 
lum breaks the circuit. Every second is thus recorded upon 
the revolving paper. The observer also holds in his hands a 
break-circuit key, with which, whenever he wishes to note the 
time, he breaks the circuit, and thus causes the pen to make its 
mark upon the paper. The line which the pen describes upon 
the paper will be something like this: 

a b c 

L i_l Li J__ 



The equidistant marks, a, b, c, etc., are the marks caused by the 
pendulum, and the marks J, B, C\ are the marks w T hich the pen 
makes when the circuit is broken by the observer. The distance 
from A to 6, B to c, etc., can be measured by a scale of equal 
parts, and the time of an observation can thus be obtained within 
a small fraction of a second. 

The cylinder is also moved by a fine screw in the direction 
of its own length, so that the pen records in a spiral. 

This instrument is called a Chronograph. There are several 
varieties of the chronograph in use by different astronomers, but 
the main principle in all of them is similar to that of the instru- 
ment just now described. 

THE TRANSIT INSTRUMENT. 

23. The transit instrument is used, as its name implies, in 
observing the transits of the heavenly bodies. Fig. 5 represents 
a transit instrument. It consists of a telescope, TT, sustained 
by an axis, AA, at right angles to it. The extremities of this 
axis terminate in cylindrical pivots, which rest in metallic sup- 
ports, VV, shaped like the upper part of the letter Y, and hence 
called the Ye. These Ys are imbedded in two stone pillars. In 
order to relieve the pivots of the friction to which the weight 
of the telescope subjects them, and to facilitate the motion of the 



32 



THE TRANSIT INSTRUMENT. 




Pig. 6. 



telescope, there are two counterpoises, WW, connected with 
levers, and acting at XX, where there are friction rollers upon 
which the axis turns. When the instrument is properly adjusted, 
the telescope, as it turns about with the axis AA, will continu- 
ally lie in the plane of the meridian; and, in order to effect this, 
the axis AA should point to the east and the west point of the 
horizon, and be parallel to its plane. There are therefore screws 
at the ends of the axis, by which one extremity of the axis may 
be raised or depressed, and may also be moved forward or 
backward. 

24. The Reticule. — In the common focus 
of the object-glass and the eye-glass is placed 
the reticule, a representation of which is given 
in Fig. 6. It consists of several equidistant 
vertical wires (usually seven) and two hori- 
zontal ones. If the instrument is accurately 
adjusted to the plane of the meridian, the 
instant that any star is on the middle wire is the instant of its 
transit. These wires are also called the cross-wires. 

25. Adjustment. — The axis of rotation of the instrument is an 
imaginary line connecting the central points of the pivots. 

The axis of coUimation is an imaginary line drawn from the 
optical centre of the object-glass, perpendicular to the axis of 
rotation. 

The line of sight is an imaginary line drawn from the optical 
centre of the object-glass to the middle wire. 

The transit instrument is accurately adjusted in the plane of 
the meridian, when the line of sight of the telescope lies con- 
tinually in that plane, as the telescope revolves. Three things, 
then, are readily seen to be necessary : the axis of rotation must 
be exactly horizontal ; it must lie exactly east and west ; and the 
line of sight and the axis of collimation must exactly coincide. 
Practically, these conditions are rarely fulfilled; but they can, by 
repeated experiments, be very nearly fulfilled, and the errors 
which the failure rigorously to adjust the instrument causes in 
'he observations will be constant and small, and can be accu- 
rately determined. 

26. Application.— The principal application of the transit 



THE MERIDIAN CIKCLE. 



33 




Fig 7. 



34 THE MERIDIAN CIRCLE, 

instrument in observatories is to the determination of the right 
ascensions of celestial bodies. Knowing the constant instru- 
mental errors just now mentioned, and the error and the rate of 
the clock, we can easily obtain the true sidereal time at the 
instant of the transit of a celestial body, which time is at once, 
as we have already seen, the right ascension of that body. 



THE MERIDIAN CIRCLE. 

27. The meridian circle is a combination of a transit instru- 
ment, similar to the one above described, and a graduated circle, 
securely fastened at right angles to the horizontal axis, and 
turning with it. A meridian circle which is set up at the United 
States Naval Academy, Annapolis, Maryland, is represented in 
Fig. 7. The horizontal axis bears two graduated circles, CC, 
C'C'y the first of these circles being much more finely graduated 
than the second, the latter being only used as a finder y to set the 
telescope approximately at any desired altitude. R and R re- 
present two of four stationary microscopes, by which the circle 
CC is read; LL is a hanging level, by which the horizontally of 
the axis is tested. The cross-wires are illuminated by light 
which passes from a lamp through the tubes A A, and through 
the pivots which are perforated for this purpose, and is reflected 
towards the reticule by a metallic speculum which is set within 
the hollow cube M. The quantity of light admitted is regulated 
by revolving discs with eccentric apertures, which are placed 
between the Ys and the tubes AA, and are moved by cords car- 
rying small weights, SS. 

The object in having so many reading microscopes is to dim- 
inish the errors arising from the imperfect graduation of the ver- 
tical circle. When any angle is to be measured, the readings of 
all four microscopes are first taken. The telescope, carrying the 
circle with it 5 is then moved through the angle whose value is 
required, and the new readings of the microscopes, which have 
remained stationary, are taken. We thus obtain four values, 
one from each microscope, of the angle measured. Theoretically, 
these values should be identical, and if they are not, their mean 
is taken as the true measure of the angle observed. 



THE MICROSCOPE. 



35 




Fig. 8. 



28. The Reading Microscope. — The reading microscope ia 
represented in Fig. 8. 
The observer, placing 
his eye at A, sees the 
image of the divisions 
of the graduated circle 
MN, formed at D, the 
common focus of the 
glasses A and C. He 
will also see a scale 
of notches, nn, and 
two intersecting spider 
threads, as shown in 
Fig. 9. These threads 
are attached to a sliding 
frame, aa, which is 
moved by means of a 
fine screw, cc, the head 
of which, EF, is gra- 
duated. The scale of 
notches is immovable, 
and is so constructed 
that the distance be- 
tween the centres of any 
two consecutive notches 
is equal to that between Fig. 9. 

the threads of the screw, thus making the number of teeth passed 
over by the spider threads equal to the number of complete re- 
volutions made by the screw. The central notch is taken as the 
point of reference, and is distinguished by a hole opposite to it. 
There is a fixed index at i, to which the divisions on the head of 
the screw are referred. When any division of the limb does not 
coincide with the central notch, the spider threads are moved 
from the central notch to the division, and the number of revo- 
lutions and fractional parts of a revolution which the screw 
makes is noted. If now w T e suppose the value of each division 
of the graduated circle, MN, to be 10', and that ten revolutions 
of the screw suffice to carry the spider threads across one of these 




86 FIXED POINTS. 

divisions, then will one revolution of the screw correspond to an 
arc of 1'; and if we further suppose that the head of the screw 
is divided into 60 equal parts, then each division on the head 
will correspond to an arc of 1". In such a case, the complete 
reading of the limb is obtained to the nearest second. By in- 
creasing the power of the microscope, the fineness of the screw, 
and the number of the graduations on the screw-head, the read- 
ing of the limb may be obtained with far greater precision. 

29. Fixed Points. — The meridian circle, being also a transit 
instrument, may be used as such; but the object for which it is 
specially used is the measurement of arcs of the meridian. In 
order to facilitate such measurement, certain fixed points of re- 
ference are determined upon the vertical circle. The most im- 
portant of these points are the horizontal point, by which is 
meant the reading of the instrument when the axis of the tele- 
scope lies in the plane of the horizon ; the polar point, which is 
the reading of the instrument when the telescope is directed to 
the elevated pole; the zenith point, and the nadir point. 

30. The Horizontal Point. — As the surface of a fluid, when at 
rest, is necessarily horizontal, and as, by the laws of Optics, the 
angles of incidence and reflection are equal to each other, the 
image of a star reflected in a basin of mercury will be depressed 
below the horizon by an angle equal to the altitude of the star 
at that instant. If, then, we take the reading of the vertical 
circle when a star w 7 hich is about to cross the meridian is on the 
first vertical thread of the reticule, and then, depressing the 
telescope, take the reading of the circle when the reflected image 
of the star crosses the last vertical thread, and, by means of 
small corrections, reduce these readings to what they would have, 
been, had both star and image been on the meridian when 
the observations were made, the mean of these two reduced 
readings will be the horizontal point. The horizontal point 
having been thus determined, the zenith point and the nadir point, 
being situated at intervals of 90° from it, are at once obtained. 
Knowing the horizontal and the zenith point, we are able to 
measure the meridian altitude or the meridian zenith distance 
of any celestial body which comes above our horizon. And 
further, as the latitude is equal to the altitude of the elevated 



NADIR POINT. 



37 



independently 





<zy 




cr 



~7D 



Fig. 10. 



pole, if the latitude of the place is accurately known, we can at 
once obtain the polar point by applying the latitude to the 
horizontal point. 

31. Nadir Point, — The nadir point may 
obtained in the following manner. Let the 
telescope, represented in Fig. 10 by AB, be 
directed vertically downwards towards a basin 
of mercury, CD. The observer, placing his 
eye at A, will see the cross-wires of the tele- 
scope, and will see also the image of these 
wires reflected into the telescope from the 
mercury. By slowly moving the telescope, 
the cross-wires and their reflected image may 
be brought into exact coincidence, and the 
reflected image will then disappear. The 
line of sight of the telescope is now vertical, 
and the reading of the vertical circle will be 
the nadir point, from which the other points can readily be found. 

There is a variety of methods by which each of these points 
can be obtained, without reference to any other; and by com- 
paring the results which these different independent methods 
give, the errors to which each result is liable may be very con- 
siderably diminished. 

32. Use of the Meridian Circle. — The meridian circle may be 
used in connection with the sidereal clock, to find the right 
ascension and declination of any celestial body. The telescope 
is directed towards the body as it crosses the meridian, and the 
time of transit as shown by the clock, and the reading of the 
vertical circle, are both taken. We have already seen how the 
right ascension of the body is obtained from the clock time of 
transit. The difference between the reading of the circle, which 
we suppose to have been taken, and the polar point, is the polar 
distance of the body, the complement of which is the declina- 
tion. Or we may obtain the declination still more directly by 
previously establishing the equinoctial point of the instrument, 
the reading, that is to say, of the vertical circle when the tele- 
scope lies in the plane of the equinoctial.* 

* A« the direction in which n star appears to lie is not, owing to refrac- 



38 



MURAL CIRCLE. 



On the other hand, if we make the same observations upon a 
star whose right ascension and declination are known, we can 
determine the latitude and the sidereal time of the place of 
observation. " 



THE MURAL CIRCLE. 

33. The mural circle is, in construction, adjustment, and use, 
essentially a meridian circle. The only important difference 
between the two instruments is in the manner in which they are 
mounted. The horizontal axis of the mural circle, instead of 
being supported at both extremities, is supported only at one, 
which is let into a stone pier or wall. Owing to the lack of 
symmetrical support, and also to the fact that the instrument 
does not admit of reversal (which is an important element in the 
adjustment of the meridian circle, and consists in lifting it out 
of the Ys, and turning the horizontal axis end for end), the 
mural circle can be regarded only as an inferior type of the 
meridian circle. 



THE ALTITUDE AND AZIMUTH INSTRUMENT. 

34. The general principles on which the altitude and azimuth 

instrument is constructed are seen 
in Fig. 11. Through the centre of 
a graduated circle, C" C", and per- 
pendicular to its plane, is passed an 
axis, AA. At right angles to this 
axis is a second axis, on$ extremity 
of which is represented by B. This 
second axis carries the telescope, TT y 
and also a second graduated circle, 
CC, whose plane is perpendicular to 
^g. ii- that of the circle C" C". The tele- 

scope admits of being moved in the plane of each circle, and 

tion and other causes, which will be explained in Chap. III., the direction 
in which it really lies, certain small corrections must be applied to the 
reading of the circle to obtain the reading which really corresponds to the 
direction of the star. 




ALTITUDE AND AZIMUTH INSTRUMENT. 



59 



microscopes or verniers are attached to the instrument, by means 
of which arcs on either circle can be read. 

If this instrument is so placed that the principal axis, AA, 
lies in a vertical direction, we shall have an altitude and azimuth 
instrument, sometimes called an altazimuth. The circle C C r 
will then lie in the plane of the horizon, and the axis AA, in- 
definitely prolonged, will meet the surface of the celestial sphere 
in the zenith and the nadir. The circle CC, as it is moved 
with the telescope about the axis AA, will continually lie in a 
vertical plane. 

35. Fixed Points. — Altitudes maybe measured on the vertical 
circle, when we know the horizontal or the zenith point, the de- 
termination of which has already been described in Arts. 30 and 
31. In order to measure the azimuth of any celestial body, we 
must, in like manner, establish some fixed point of reference on 
the horizontal circle, as, for instance, the north or the south 
point, by which is meant the reading of the horizontal circle 
when the telescope lies in the plane of the meridian. 

36. Method of Equal Altitudes. — One of the most accurate 
methods of obtaining the north or the 

south point of the horizontal circle is 
called the method of equal altitudes. 
Let Fig. 12 represent the projection of 
the celestial sphere on the plane of the 
celestial horizon, NESW. Z is the 
projection of the zenith, P of the pole, 
and the arc AA! the projection of a 
portion of the diurnal circle of a fixed 
star, which is supposed to have the 
same altitude when it reaches A', west 
of the meridian, which it had at A, east of the meridian. 

ISTow, in the two triangles PAZ, PA' Z, we have PZ common, 
PA' equal to PA (since the polar distance of a fixed star re- 
mains constant), and ZA equal to ZA\ by hypothesis ; the two 
triangles are therefore equal in all their parts, and hence the 
angles PZA and PZA f are equal. But these two angles are the 
azimuths of the star at the two positions A and A'. We may 




40 EQUATORIAL. 

say, then, in general, that equal altitudes of a fixed star corre- 
spond to equal angular distances from the meridian. 

Now, let the telescope be directed to some fixed star east of the 
meridian, and let the reading of the horizontal circle be taken. 
When the star is at the same altitude, west of the meridian, let 
the reading of the horizontal circle again be taken ; the mean of 
these two readings is the reading of the horizontal circle when 
the axis of the telescope lies in the plane of the meridian. 

37. Use of the Altitude and Azimuth Instrument — This instru- 
ment is chiefly used for the determination of the amount of re- 
fraction corresponding to different altitudes. Refraction, as will 
be seen in the next chapter, displaces every celestial body in a 
vertical direction, making its apparent zenith distance less than 
its true zenith distance. At the instant of taking the altitude 
of a celestial body, the local sidereal time is noted, from which, 
knowing the right ascension of the body, w T e can obtain its hour 
angle from the theorem in Art. 9. We shall then have in the 
astronomical triangle, PZA (Fig. 3), the hour angle ZPA, the 
side PA, or the polar distance of the body, and the side PZ, the 
co-latitude of the place of observation. We can, therefore, com- 
pute the side ZA, which is the true zenith distance of the body 
observed ; and the difference between this and the observed zenith 
distance, (corrected for parallel, [Art. 55,] and instrumental 
errors,) will be the amount of refraction for that zenith distance. 

The construction of the instrument enables us to follow a 
celestial body through its whole course from rising to setting, 
measuring altitudes and noting the corresponding times to any 
extent that we choose; and the amount of refraction correspond- 
ing to each altitude can afterwards be computed at our leisure. 



THE EQUATORIAL. 

38. The equatorial is similar in general construction to the 
altitude and azimuth instrument. It is, however, differently 
placed, the plane of the principal graduated circle, C C", coin- 
ciding, not with the plane of the horizon, but with that of the 
celestial equator, from which peculiarity of position comes the 



EQUATOIUAL. 41 

name of equatorial. The circle C C\ when thus placed, is called 
the hour circle of the instrument, and the axis J., at right angles 
to it, is called the hour or polar axis. It is evident that the axis 
is directed towards the poles of the heavens. The circle CC\ 
the plane of which is perpendicular to the plane of the circle 
C'C, will lie continually in the plane of a circle of declination, 
as the instrument is turned about the polar axis, and is hence 
called the declination circle. The axis on which this latter circle 
is mounted is called the declination axis. 

39. Use of the Equatorial. — The equatorial is employed prin- 
cipally in that class of observations which require a celestial 
body to remain in the field of view during a considerable length 
of time. The manner in which this 
requirement is met is explained /O 

in Fig. 13. Let A A' be the polar / * 

axis of an equatorial, directed to- #>/ / 

wards the pole of the heavens, P. / \£ 

Let sss f be the diurnal circle in / / x %. 
which a star appears to move / / v \ T 

about the pole. Suppose the tele- +■ ( > NL ID 3 

scope, TT, to be turned in the di- a 

rection of the star when at s, and Fig. is. 

to be moved until the intersection of the cross-wires and the star 
coincide, and then clamped. Now, if the instrument is made to 
revolve about the axis AA\ with an angular velocity equal to 
that of the star about the pole, it is plain that, since the angle 
which the axis of the telescope makes with the polar axis remains 
unchanged, and is continually equal to the angular distance of 
the star from the pole, the coincidence of the cross-wires and the 
star will remain complete. 

A clock-work arrangement, called a driving-clock, is now usually 
connected with large equatorials, by which the instrument may 
be moved uniformly about its polar axis, at the required rate, so 
that the observer has ample time to measure the angular diameter 
of a celestial body, to measure the angular distance between two 
stars which are near each other, and to make other micrometric 
observations of a similar character. 



42 



SEXTANT. 



THE SEXTANT. 

40. The sextant is an instrument by which the angular dis- 
tance between two visible objects may be measured. It is used 
chiefly by navigators ; but its portability gives it great value 
wherever celestial observations are required. The angles for 
the measurement of which it is used are the altitudes of celestial 
bodies, and trie angular distances between celestial bodies or 
terrestrial objects. 

Fig. 14 is a representation of the sextant. Its form is that 




of a sector of a circle, the arc of which comprises 60°. A movable 
arm, CD, called the index-bar, revolves about the centre of the 
sector. This bar carries at one extremity a vernier, Z>. At the 
other extremity of the index-bar, and revolving with it, is placed 
a silvered mirror, C, the surface of which must be perpendicular 
to the plane of the instrument. This glass is called the index- 
glass. Another glass, N, called the horizon- glass, is attached to 
the frame of the instrument, and only its lower half is silvered. 



SEXTANT. 43 

This glass is immovable, and its surface must be perpendicular 
to the plane of the instrument. T is a telescope, directed to- 
wards the horizon-glass, with its line of sight parallel to the 
plane of the instrument. F and E are two sets of colored 
glasses, which may be used to protect the eye when the sun is 
observed. M is a magnifying glass, to assist the eye in reading 
the vernier. G is a tangent screw, which gives a slight motion to 
the index-bar, and is used in obtaining an accurate coincidence 
of the images. 

41. Optical Principle of Construction.— The sextant is con- 
structed upon a principle in Optics which may be stated thus : — 
The angle between the first and the last direction of a ray which 
has suffered two reflections in the same plane is equal to twice the 
angle which the two reflecting surfaces make with each other. 

To prove this: In Fig 15, 
let A and jB be the two re- \s 

fleeting surfaces, supposed to \ 

be placed with their planes \ 

perpendicular to the plane of ' " ^«..^ \ t 

the paper. Let SA be a ray //l'^- 

of light from some body, S, yS \ "~J>-c 

which is reflected from A to .._ \/^— X" 

B, and from I> in the direction Jl" ! -. _ \ 

BE. Prol on gSA until it meets F 15 

the line BE. Then will the 

angle SEB be the angle between the first and the last direction 
of the ray SA. At the points A and B let the lines AD and 
BChe drawn perpendicular to the reflecting surfaces, and pro- 
long AD until it meets BC The angle DCB is equal to the 
angle which the tw r o surfaces make with each other. We have 
then to prove that the angle SEB is double the angle DCB. 

Now since the angle of incidence always equals the angle of 
reflection, SAD and DAB are equal, and so are ABC and CBE. 
We have, by Geometry, 

SEB = SAB — ABE 

= 2 {DAB — ABC), 
= 2 DCB. 

42. Measurement of Angular Distances. — Suppose, now, that 



44 



SEXTANT. 









a\. 


a' 

k 




B' 


















\ 




& 


0< S 


rrp\ 


' 




\i 



Fig. 16. 



we wish to measure the angular distance between two celestial 
bodies, A and B (Fig. 16). The instrument is so held that its 
plane passes through the two bodies, and the fainter of them, 

which in this case we suppose to 
be B y is seen directly through the 
horizon-glass and the telescope. 
B is so distant that the rays B' C 
and Bm } coming from it, may be 
considered to be sensibly parallel. 
Let ab and CI be the positions of 
the index-glass and index-bar when 
~D the index-glass and the horizon- 
glass are parallel. Then will the 
ray B'C be reflected by the two 
glasses in a direction parallel to 
itself, and the observer, whose eye is 
at D, will see both the direct and the 
reflected image of B in coincidence. Now let the index-bar be 
moved to some new position, CI', so that the ray from the second 
body, A y shall be finally reflected in the direction of mD. The 
observer will then see the direct image of B and the reflected 
image of A in coincidence; and the angular distance between 
the two bodies is evidently equal to the angle between the first 
and the last direction of the ray A (7, which angle has already 
been shown to be equal to twice the angle which the two glasses 
now make with each other, or to twice the angle ICI f . If, then, 
we know the point Jon the graduated arc at which the index-bar 
stands when the glasses are parallel, twice the difference between 
the reading of that point and that of the point I' will be the 
angular distance of the two bodies. 

To avoid this doubling of the angle, every half degree of the 
arc is marked as a whole degree, when the graduation is made; 
so that, in practice, we have only to subtract the reading of / 
from that of T to obtain the angle required. 

43. Index Correction.— The point of reference on the arc from 
which all angles are to be reckoned is, as we have already seen, 
the reading of the sextant when the surfaces of the index-glass 
and the horizon-glass are parallel. This point may fall either 



ARTIFICIAL HOKIZOX. 45 

at the zero of the graduation, or to the left or to the right of it; 
and to provide for the last case, the graduation is carried a short 
distance to the right of the zero, this portion of the arc being 
called the extra arc. The reading of this point of parallelism 
is called the index correction, and is positive when it falls to the 
right of the zero, and negative when it falls to the left. Suppose, 
for instance, that the instrument reads 2' on the extra arc when 
the glasses are parallel: all angles ought then to be reckoned 
from the point 2', instead of from the zero point ; in other words, 
2' is a constant correction to be added to every reading. 

There are several methods of finding the index correction. 
One method, which can readily be shown from Fig. 16 to be a 
legitimate one, is to move the index-bar until the direct and the 
reflected image of the same star are in coincidence, and then 
take the reading, giving it its proper sign according to the rule 
above stated. Another method, generally more convenient, in 
which the sun is used, may be found in Bowditch's Navigator, 
and in most treatises on Astronomy: w T here also may be found 
the methods of testing the adjustments of the sextant. 

44. The Artificial Horizon. — In order to obtain the altitude 
of a celestial body at sea, the sextant is held in a vertical posi- 
tion, and the index-bar is moved until the reflected image of the 
body is brought into contact with the visible horizon seen through 
the telescope of the sextant. The sextant reading is then cor- 
rected for the index correction; and corrections must also be 
applied for parallax, refraction, and the dip of the horizon, as 
will be explained in the next Chapter. If the body observed is 
the sun or the moon, either its upper or its lower limb is brought 
into contact with the horizon, and the value of its angular 
semi-diameter (given in the Nautical Almanac) is subtracted or 
added. 

On shore, use is made of the artificial horizon, already alluded 
to in Art. 30. This commonly consists of a shallow, rectangular 
basin of mercury, the surface of which is protected from the 
wind by a sloping roof of glass. The observer so places himself 
that he can see the image of the body whose altitude he wishes 
to measure reflected in the mercury. He then moves the index- 
bar of the sextant until the image of the body reflected by the 



4(5 



VERNIER. 



sextant is in coincidence with that reflected by the mercury. 
The sextant reading is then corrected for the whole of the index 
correction. Half of the result will be, as shown in Art. 30, 
the apparent altitude of the body, to which must be applied 
the corrections for parallax and refraction to obtain the true 
altitude. When the sun or the moon is observed, the upper or 
the lower limb of the image reflected by the sextant is brought 
into contact with the opposite limb of the image reflected by the 
mercury, and the correction for semi-diameter also is applied. 

45. The Vernier. — The vernier is an instrument by which, as 
by the reading microscope previously explained, fractions of a 

division of a limb may be read. 
In Fig. 17, let AB be an arc of 
a stationary graduated circle, 
and let CD be a movable arm, 
carrying another graduated 
arc at its extremity. The value 
of each division of the limb 
AB is one-sixth of a degree, or 
10'. The arc on the arm CD 
is divided into ten equal parts, 
and the length of the arc be- 
~i2° tween the points and 10 is 

Fi s- 17 - equal to the length of nine di- 

visions of the arc AB. This arc, which the limb CD carries, is 
called a vernier. Since the ten divisions of the vernier equal in 
length nine divisions of the limb, it follows that each division 
of the vernier comprises 9' of arc ; in other w T ords, any division 
of the vernier is less by V of arc than any division of the limb. 
The reading of any instrument which carries a vernier is al- 
ways determined by the position of the zero point of the vernier. 
If, now, the zero point of the vernier exactly coincides with a 
division of the limb, the point 1 of the vernier will fall V behind 
the next division of the limb, the point 2 will fall 2' behind the 
next division but one, and so on; and if, such being the case, the 
vernier is moved forward through an arc of 1', the point 1 will 
come into coincidence with a division of the limb; if it is moved 
forward through an arc of 2', the point 2 will come into coinci- 




VERNIER. 47 

dence with a division on the limb ; and, in general, the number 
of minutes of arc by which the zero point of the vernier falls 
beyond the division of the limb which immediately precedes it 
will be equal to the number of that point of the vernier which 
is in coincidence with a division of the limb. If, then, the zero 
point falls between any two divisions of the limb, as 11° 20' and 
11° SO', for example, and the point 2 of the vernier is found to 
be in coincidence with any division of the limb, we know that 
the zero point is 2' beyond the division 11° 20', and that the com- 
plete reading for that position of the vernier is 11° 22'. 

46. General Rides of Construction. — In the construction of all 
verniers similar to the one above described, the same rules of 
construction must be followed : the length of the arc of the ver- 
nier must be exactly equal to the length of a certain number 
(no matter what) of the divisions of the limb, and the arc must 
be divided into equal parts, the number of which shall be greater 
by one than the number of these divisions of the limb. Following 
these rules, and putting 
J) = the value of a division of the limb, 
d = " " " " " vernier, 
n = the number of equal parts into which the vernier is divided, 

we have D — d = — as a general formula. 

n & 

The difference D — cZ is called the least count of the vernier. 

If, in Fig. 17, we take the length of the vernier equal to 59 

divisions of the limb, and divide it into 60 equal parts, we shall 

have 

which is the least count on most of the modern sextants. 

Verniers are sometimes constructed in which the number of 
equal parts on the vernier is less by one than the number of the 

D 

divisions of the limb taken. In this case we have d — D = — • 

IV 

and the only difference between this class of verniers and the 
class above described is that the graduations of the limb and 
the vernier proceed in this class in opposite directions. 



48 SPECTR08C0FE. 



OTHER ASTRONOMICAL INSTRUMENTS. 

47. The zenith telescope, the theodolite, and the universal instru- 
ment are, in general principle, only modified forms of the port- 
able altitude and azimuth instrument. 

The octant (sometimes improperly called the quadrant) is 
identical in construction with the sextant, excepting only that 
its arc contains 45°. 

The prismatic sextant carries a reflecting prism in place of the 
ordinary horizon-glass, and the graduated arc comprises a semi- 
circumference. 

The reflecting circle is still another modification of the sex- 
tant, in which the graduated arc is an entire circumference, and 
the index-bar is a diameter of the circle, revolving about the 
centre, and carrying a vernier at each extremity. Sometimes 
the circle has three verniers, at intervals of 120° of the gradu- 
ated arc. 

The spectroscope is an instrument which is used, as its name 
indicates, in the examination of the spectra both of terrestrial 
substances and of the heavenly bodies. Its use as an instrument 
of astronomical research is comparatively recent, but it has 
already led to many interesting and remarkable discoveries con- 
cerning the constitution of the heavenly bodies. It consists 
essentially of three parts: a tube, a prism (or a set of prisms), 
and a telescope. Kays of light from either a celestial body or 
an artificial flame are made to enter the tube through an ex- 
tremely narrow slit at its extremity. These rays pass through 
the tube, and fall upon the prism. If necessary, lenses may be 
placed within the tube, so that the rays, as they issue from it, 
shall fall upon the prism in parallel lines. These rays are dis- 
persed by the prism, and a spectrum is formed. This spectrum 
is then examined by means of the telescope. There is also an 
arrangement by which rays of light from two substances or 
bodies can be introduced through the slit without interfering 
with each other, so that their spectra can be formed simul- 
taneously, one above the other, and the points of resemblance or 
difference between them can be accurately noted. 

It is well known that the solar spectrum contains a large 



ERRORS. 49 

number of dark and narrow parallel lines, which are called 
Fraunhofer's lines. The spectra of the stars and of artificial 
lights also contain similar series of lines, differing from each 
other, each series, however, being constant for the same body or 
light. The spectra of chemical substances also present certain 
peculiarities, so that each spectrum indicates with certainty the 
substance which produces it. Hence, by a comparison of the 
spectra of the heavenly bodies with those of known chemical 
substances, the existence of many of those substances in the 
heavenly bodies has been definitely established. Nor is this all ; 
the inspection of any spectrum suffices to tell us whether the 
light which forms it comes from a solid or a gaseous body, and 
whether, if the light comes from a solid body, it passes through 
a gaseous body before it reaches us. 

The results of these investigations will be noticed when we 
come to the description of the heavenly bodies ; and the method 
of investigation will be further illustrated in the Article on the 
constitution of the sun. (Art. 102.) 



ERRORS. 

48. However carefully an instrument may be constructed, how- 
ever accurately adjusted, and however expert the observer may 
be, every observation must still be regarded as subject to errors. 
These errors may be divided into two classes, regular and irre- 
gular errors. By regular errors we mean errors which remain 
the same under the same combination of circumstances, and 
which, therefore, follow some determinate law, which may be 
made the subject of investigation. Among the most important 
of this class of errors are instrumental errors: errors, that is to 
say, due to some defect in the construction or adjustment of an 
instrument. If, for instance, what we call the vertical circle of 
the meridian circle is not rigorously a circle, or is imperfectly 
graduated; or if the horizontal axis is not exactly horizontal, 
or does not lie precisely east and west ; any one of these imper- 
fections will affect the accuracy of the observation. The observer, 
however, knowing what the construction and adjustment of the 
instrument ought to be, can calculate what effect any given im- 



50 ERRORS. 

perfection will produce upon his observation, and can thus de- 
termine what the observation would have been had the imper- 
fection not existed. Regular errors, then, may be neutralized 
by determining and applying the proper corrections. 

Irregular errors, on the contrary, are errors which are not 
subject to any know T n law. Such, for example, are errors pro- 
duced in the amount of refraction by anomalous conditions of 
the atmosphere ; errors produced by the anomalous contraction 
or expansion of certain parts of the instrument, or by an un- 
steadiness of the telescope produced by the wind ; and, more par- 
ticularly, errors arising from some imperfection in the eye or the 
touch of the observer. Errors such as these, being governed 
by no known law T , can never be made the subject of theoretic 
investigation; but being by their very nature accidental, the 
effects which they produce will sometimes lie in one direction 
and sometimes in another ; and hence the observer, by repeating 
his observations, by changing the circumstances under which he 
makes them, by avoiding unfavorable conditions, and finally by 
taking the mean, or the most probable value of the results which 
his different observations give him, can very much diminish the 
errors to which any single observation would be exposed. 

Note. — For complete descriptions of the various astronomical instru- 
ments, the student is referred to Chauvenet's Spherical and Practical As- 
tronomy; Loomis's Practical Astronomy ; and Pearson's Practical Astronomy 
(published in England). 



REFRACTION. 51 



CHAPTER III. 

REFRACTION. PARALLAX. DIP OF THE HORIZON. 
REFRACTION. 

49. When a ray of light passes obliquely from one medium 
to another of different density, it is bent, or refracted, from its 
course. If a line is drawn perpendicular to the surface of the 
second medium at the point where the ray meets it, the ray is 
bent towards this perpendicular if the second medium is the 
denser of the two, and from it if the first medium is the 
denser. 

In Fig. 18, let AA, BB, represent two 
media of different density, the density of 
BB being the greater. Let CD be a 
ray of light meeting the surface of BB 
at D. At D erect the line ND perpendi- 
cular to the surface of BB, and prolong g ~m~ 
it in the direction DM. The ray CD is Fi ^ 18 - 
called the incident ray, and the angle NDCihe angle of inci- 
dence. When the ray enters the medium BB, it will still lie in 
the same plane with CD and ND, but will be bent towards the 
line DM, making with it some angle GDM, less than the angle 
NDC. To an observer whose eye is at G, the ray will appear 
to have come in the direction EG, which is therefore called the 
apparent direction of the ray. DG is called the reflected -ray, 
and the angle GD3I the angle of refraction. The angle EDC, 
the difference between the directions of the incident and the 
refracted ray, is called the refraction. 

It is shown in Optics that, whatever the angle of incidence 
may be, there always exists a constant ratio between the sine 
of the angle of incidence and that of the angle of refraction, 
as long as the same two media are used and their densities are 
unchanged. We have, then, in the figure, — 



N E 


C 


A 


y< 


/ 


D 




- / 




B 



52 



REFRACTION. 



sin NDC 

sin GDM~~ ' 

k being a constant for the two media AA and BB. 

If the second medium, instead of being of uniform density, is 
composed of parallel strata, each one of which is of greater 
density than the one immediately preceding, as is represented 

in Fig. 19, the path of the ray through 
these several strata will be a broken 
line, Dabc; and if the thickness of 
B ft> b each of these successive strata is sup- 

i c posed to be indefinitely small, this 

Fi «- 1J - broken line will become a curve. 

In the figures above used, the media are represented as sepa- 
rated by plane surfaces; but the same phenomena are noticed, 
and the same laws hold good, if the media are separated by 
curved surfaces. 

50. Astronomical Be fraction. — It is determined by experiment 
that the density of the air gradually diminishes as we ascend 
above the surface of the earth, and it is estimated that at a 
distance of fifty miles above the surface the upper limit of the 
air is reached; or, at all events, that the density of the air is so 
small at that distance that it exerts no appreciable refracting 
power. We may, therefore, consider the air to be made up of 
a series of strata concentric with the earth's surface, the thick- 
ness of each stratum being in- 
definitely small, and the den- 
sity of each stratum being 
greater than that of the stra- 
tum next above it. Now, in 
Fig. 20, let the arc BB repre- 
sent a portion of the earth's 
surface, and the arc 31N a por- 
tion of the upper limit of the 
atmosphere. Let S be a ce- 
lestial body, and SA a ray of 
light from it, which enters the 
atmosphere at A. Let the 
Fig. 20. normal (or radius) A C be 




REFRACTION. 53 

drawn. As the ray of light passes down through the atmev 
phere, it is continually passing from a rarer to a denser 
medium, so that its path is continually changed, and becomes 
a curve AL, concave towards the earth, and reaching the earth 
at some point L. Since the direction of a curve at any point 
is the direction of the tangent to the curve at that point, the 
apparent direction of the ray of light at L will be repre- 
sented by the tangent LS\ and in that direction will the body 
S appear to lie, to an observer at L. If the radius CL be 
indefinitely prolonged, the point Z, where it reaches the celestial 
sphere, will be the zenith of the observer at L, the angle ZLS 
will be the apparent zenith distance of the body 8, and th •- 
angle which the line drawn from the body to the point 7> 
makes with the line LZ will be the true zenith distance of S. 
The effect, then, of refraction is to decrease the apparent zenith 
distances, or increase the apparent altitudes of the celestial 
bodies. Since the incident ray SA, the curve AL, and the 
tangent LS' all lie in the same vertical plane, the azimuth of the 
celestial bodies is not affected. 

51. General Laics of Refraction. — By an investigation of the 
formulae of refraction, and by astronomical observations already 
described (Art. 37), the amount of refraction at different alti- 
tudes has been obtained, and is given in what are called "tables 
of refraction." The following general laws of refraction will 
serve to give the student some idea of its amount, and of the 
conditions under which it varies: — 

(1.) In the zenith there is no refraction. 

(2.) The refraction is at its maximum in the horizon, being 
there equal to about 33'. At an altitude of 45° it amount* 
to 57". 

(3.) For zenith distances which are not very large, the re- 
fraction is nearly proportional to the tangent of the zenith 
distance. When the zenith distance is large, however, the ex- 
pression of the law^ is much more complicated. No table of re- 
fraction can be trusted for an altitude of less than 5°. 

(4.) The amount of refraction depends upon the density of the 
air, and is nearly proportional to it. The tables give the re- 
fraction for a mean state of the atmosphere, taken with the 



54 PARALLAX. 

barometer at 30 inches and the thermometer at 50°. If the 
temperature remains constant, and the barometer stands above 
its mean height, or if the height of the barometer is constant, 
and the thermometer stands below its mean height, the density 
of the atmosphere is increased, and the refraction is greater 
than its mean amount. Supplementary tables are therefore 
given, from which, with the observed heights of both barometer 
and thermometer as arguments, we may take the necessary cor- 
rections to be applied to the mean refraction. 

(5.) Since the effect of refraction is to increase the apparent 
altitudes of the celestial bodies, the amount of refraction for 
any apparent altitude is to be subtracted from that apparent 
altitude, or added to the corresponding zenith distance. 

52. Effects of Refraction. — The apparent angular diameter of 
the sun and of the moon being about 32', and the refraction in 
the horizon being 33', it follows that when the low T er limb of 
either body appears to be resting on the horizon, the body is in 
reality below T it. One effect, then, of refraction is to lengthen 
the time during which these bodies are visible. Still another 
effect is to distort the discs of the sun and the moon when near 
the horizon : for since the refraction varies rapidly near the 
horizon, the lower extremity of the vertical diameter of the 
body will be more raised than the upper extremity, thus appa- 
rently shortening this diameter, and giving the body an ellip- 
tical shape. When the body comes still nearer the horizon, its 
disc is distorted into what is neither a circle nor an ellipse, but 
a species of oval, in which the curvature of the lower limb is 
less than that of the upper one. The apparent enlargement of 
these bodies when near the horizon is merely an optical delu- 
sion, which vanishes when their diameters are measured with an 
instrument. 

PARALLAX. 

53. The parallax of any object is, in the general sense of the 
word, the difference of the directions of the straight lines drawn 
to that object from two different points: or it is the angle at 
the object subtended by the straight line connecting these two 
points. In Astronomy, we consider two kinds of parallax : yeo- 



PARALLAX. 



55 



centric parallax, by which is meant the difference of the directions 
of the straight lines drawn to the centre of any celestial body from 
the earth's centre and any point on its surface, and heliocentric 
parallax, or the difference of the directions of the lines drawn to 
the centre of the body from the cen- 
tre of the earth and the centre of the 
sun. The former is the angle at the 
body subtended by that radius of 
the earth which passes through the 
place of observation : the latter the 
angle at the body subtended by the 
straight line joining the centre of 
the earth and that of the sun. 

54. Geocentric Parallax. — In Fig. 
21, let Cbe the centre of the earth, 
and L some point on its surface, of which Z is the zenith. Let 
S be some celestial body. The geocentric parallax of the body 
is the angle CSL. Let S' be the same body in the horizon. 
The angle LS' C is the parallax of the body for that position, 
and is called its horizontal parallax. If we denote this horizontal 
parallax by P, the earth's radius by P, and the distance of the 
body from the earth's centre by d, we have, by Trigonometry, 

R 




Fig. 21. 



sin P = 



d 



To find the parallax for any other position, as at S, we repre- 
sent the angle LSC by p, and the apparent zenith distance 
of the body, or the angle ZLS, by z, the sine of which is equal 
to the sine of its supplement SLC. We have from the tri- 
angle LSC, since the sides of a plane triangle are proportional 
to the sines of their opposite angles, 

sin p R 

sin z d 
Combining this equation w T ith the preceding, w T e have, 

sin p = sin Psin z. 
Since P and p are small angles, we may consider them propor- 
tional to their sines, and thus have, finally, 

p — P sin z. 
The parallax, then, is proportional to the sine of the zenith 



06 PARALLAX. 

distance, and may be found for any altitude when the hori- 
zontal parallax is known. It evidently decreases as the alti- 
tude increases, and in the zenith becomes zero. 

55. Application of Parallax. — In order that observations 
made at different points of the earth's surface may be com- 
pared, they must be reduced to some common point. Geocen- 
tric parallax is applied to reduce any altitude observed at any 
place to what it would have been had it been observed at the 
earth's centre. We see from Fig. 21 that parallax acts in a 
vertical plane, and that the zenith distance of the body as ob- 
served from the earth's centre, or the angle ZCS, is less than the 
observed zenith distance ZLS, by the parallax CSL. Parallax, 
then, is always subtractive from the observed zenith distance, 
and additive to the observed altitude. 

The parallax above described is, strictly speaking, the paral- 
lax in altitude. There is also, in general, a similar parallax in 
right ascension, and in declination, formulae for deriving which 
from the parallax in altitude are given in other works. 

56. Heliocentric Parallax. — It may sometimes happen that we 
wish to reduce an observation from what it was at the centre 
of the earth to what it would have been if it had been made 
at the centre of the sun. Fig. 21, and the formulae obtained 
from it, will apply equally well to this case, by making the 
necessary changes in the description of the figure and in the 
names of the angles. 

Let S be still a celestial body, but let C be the centre of the 
sun, and L that of the earth. The angle p will then represent 
the heliocentric parallax, and the angle SLC the angular dis- 
tance of the body from the sun, as measured from the earth's 
centre, or, as it is called, the body's elongation. The angle P 
will be the greatest value of the heliocentric parallax, taken 
when the body's elongation from the sun is 90°, and is called 
the annual parallax. We shall then find, from the formula 
of Art. 54, that the annual parallax has for its sine the ratio 
of the distance of the earth from the sun to that of the body 
from the sun, and that the parallax for any other position is 
the product of the annual parallax by the sine of the body's 
elongation. 



DIP OF THE HOPIZOX. 



57 



DIP OF THE HOPvIZON. 

57. The dip of the horizon is the angular depression of the 
visible horizon below the celestial hori- ^_ 
zon. In Fig. 22, let HG be a por- 
tion of the earth's surface/ and C 
the earth's centre. Let a radius of 
the earth, CA, be prolonged to some 
point D, beyond the surface, and let 
an observer be supposed to be at the 
point D. At the point D let the line 
BD be drawn perpendicular to the line 
CD, and also the line DH, tangent to 



Fig. 22. 



the earth's surface at some point H. If these two lines be revolved 
about the line CD, DB will generate the plane of the celestial 
horizon (since we have seen that all planes passed perpendicular 
to the radius will, when indefinitely extended, mark out the 
same great circle on the celestial sphere), and DH will gene- 
rate the surface of a cone, which will touch the earth in a 
small circle. If we disregard for the present the effect of the 
earth's atmosphere, this small circle will be the visible horizon 
of the observer at D, and the angle BDH will be the dip. DA 
is the linear height of the observer at D. 

Now let a radius, CS, be drawn to the point of tangency H. 
The angles BDH and HCD, having their sides mutually per- 
pendicular, are equal. Represent the angle HCD by D, the 
earth's radius by R, and the observer's height, AD, by h. In 
the triangle DHC, right-angled at H, we have, 

== COS 1) ' 

since D is small, it is better to use the formula, 

cos D = 1 — 2 sin 2 i D ; 
hence we have, 



R 



R 



-h 
I D 



\ 2{R + h). 



oS 



DIP OF THE HORIZON. 



As the angle D is small, we may take 

sin } D = i D sin 1" ; 
and as h is very small in comparison with R, we may also 
assume (jR-f-A) to be sensibly equal to R. Making these 
changes in the above equation, and finding the value of D y we 
have 

2 



D = 



sin 1' 



v 



h 
2R 



Substituting in this expression the value of the earth's radius 
in feet, we have, finally, 

D = 63".8 j/X" 
h being expressed in feet. 

The dip, then, for a height of one foot is 63".8; and for 
other heights it is proportional to the square root of the number 
of feet in the height. 

58. Effect of Atmospheric Refraction. 
— If the effect of atmospheric refrac- 
tion is taken into consideration, the 
line HD must be a curved line, as is 
represented in Fig. 23. The point H 
will then appear to lie in the direc- 
tion DH', to the observer at D, and 
the dip will be the angle BDH f . The 
Fi s- 23 -* effect, then, of refraction is to decrease 

the dip, the amount by which it is decreased being about y^th 
of the whole. 

59. Application of Dip. — Tables have been computed in 
which may be found the proper dip for different heights above 
the surface of the earth. Dip constitutes one of the correc- 
tions which are to be applied at sea to the observed altitude of 
a celestial body to obtain its true altitude: its altitude, that is, 
above the celestial horizon. Since the visible horizon lies below 
the celestial horizon, this correction is evidently subtractive. 




* The curved line HD'is tangent to the earth's surface at H. 



MEASUREMENT OF THE EARTH. 59 



CHAPTER IV. 

THE EARTH. ITS SIZE, FORM, AND ROTATION. 

60. Haying seen what are the construction and the adjustments 
of the principal astronomical instruments, to what uses each is 
adapted, and what corrections are to be applied to the observa- 
tions which are taken, we are now ready to proceed to the solu- 
tion of some of the many questions of interest w T hich the study 
of Astronomy opens to us. And first of all, let us see how as- 
tronomical observations will help us to a knowledge of the size 
and the form of the earth. The question is one of the first im- 
portance; for upon the determination of the size of the earth 
depends in a great measure, as we shall see further on, the deter- 
mination of the magnitudes and the distances of the other 
heavenly bodies. Having seen the facts which seem to point to 
the conclusion that the earth is spherical in form, we will start 
w T ith the assumption that this conclusion is correct, and proceed 
to determine the magnitude of the earth, regarded as a sphere. 

Now, we know T from Geometry what is the ratio between the 
radius of a sphere and the circumference of any great circle of 
that sphere ; and, therefore, if we can obtain the length of the 
circumference of any great circle of the earth, of a meridian, 
for instance, we can at once determine the radius of the earth. 
And more than this : if we can measure the length of any known 
arc of this meridian, of one degree, for instance, we can compute 
the length of the entire circumference. The determination of 
the earth's radius, then, depends only on our ability to satisfy 
these two conditions : 

(1.) We must be able to measure the linear distance on the 
earth's surface between two points on the same meridian. 

(2.) We must be able to measure the angular distance between 
these same two points. 

61. First Condition. — A reference to Fig. 21 will explain how 



60 



MEASUREMENT OF THE EARTH. 




Fig. 24. 



the first of these two conditions may be satisfied. Let A and G 
represent the two points on the same meridian the 
distance between which we wish to measure. We 
have already seen (Art. 36) how an altitude and 
azimuth instrument may be so adjusted that the 
sight-line of the telescope will lie in the plane of 
the meridian. Let an instrument be so adjusted 
at the point A. Let some convenient station B be 
taken, visible at A, and let the distance AB be 
carefully measured. This distance is called the 
base-line. Now, by means of the telescope, adjusted 
to the plane of the meridian, let some point C be 
established on the meridian, which shall also be 
visible from B, and let the angles CAB and ABC 
be measured. We now know in the triangle ABC 
two angles and the included side, and can compute the distances 
AC and CB. We now take some suitable point D, measure 
the angles DCB and DBC, and knowing CB, we can obtain the 
distance CD. The instrument is now taken to the point C, and 
again established in the plane of the meridian, either by the 
method of Art. 36, or by sighting back, as it is called, to A, or by 
both methods combined. A third point on the meridian, E, 
visible from D, is then selected, the angles ECD and ED C are 
measured, and the distances EC and ED computed. This pro- 
cess is continued until the whole distance between A and G has 
been obtained. 

This method of measurement is called the method of triangu- 
lation. The base-line, AB, is purposely taken under circum- 
stances which favor its accurate measurement, and the rest of 
the work consists in the determination of horizontal angles, 
which presents no special difficulty, and in the solution of tri- 
angles by computation. 

Two things are to be noticed in reference to these triangles. 
The first is that in selecting the points B, C, D, &c, care must be 
taken so to choose them that the triangles ABC, BCD, &c, shall 
be nearly equiangular; since triangles in which there is a great 
inequality of the angles {ill-conditioned triangles, as they are 
called) will be much more likely to cause some error in the work. 



MEASUREMENT OF THE EARTH. 61 

The second thing to be noticed is that these triangles are really 
spherical triangles, and must therefore be solved by the formulae 
of Spherical Trigonometry. If, for any reason, we are forced 
to take any of the points C, E, &c, off the meridian, the cor- 
responding distances can be reduced to the meridian by appro- 
priate formulae. 

The correctness of the result may be tested by measuring the 
distance GF, and comparing its measured length with that ob- 
tained by computation. The marvellous accuracy of this method 
of measurement is shown by the fact that in an arc of the meri- 
dian measured by the French at the close of the last century, 
and which was several hundred miles in length, the discrepancy 
between the measured and the computed length of the second 
base-line was less than twelve inches. 

62. Second Condition, — The second condition requires that 
the angle at the centre of the earth, subtended by the arc of the 
meridian measured, shall be obtained. This angle is evidently 
the difference of latitude of the two extremities of the arc, and 
therefore all that is needed to satisfy this condition is that the 
latitude of each extremity shall be determined by appropriate 
observations. 

Instead of determining the latitude of each place independently 
of the other, we may, if we choose, obtain the difference of 
latitude directly, by observing at each place the meridian zenith 
distance of the same celestial body. In Fig. 25, let A and G be 
the two extremities of the arc, C the centre 
of the earth, and S the celestial body on 
the meridian. If Z' is the zenith of the 
point J., the meridian zenith distance of S 
at A, reduced to the centre of the earth, is 
the angle Z' CS. In the same manner the 
true meridian zenith distance of S at G is 
the angle ZCS. The difference of these two 
zenith distances, or the angle ZCZ r , is evi- 
dently the difference of latitude of G and A. 

If the celestial body crosses the meridian between the two 
zeniths, as at $' ', the difference of latitude is numerically the 
sum of the two meridian zenith distances. 




62 FORM OF THE EARTH. 

63. Results. — By the process above described, or by processes 
of a similar character, arcs of different meridians, and in differ- 
ent latitudes, have been carefully measured. The sum of the 
arcs thus measured is more than 60°, and the length of a degree 
of the meridian has been found to be, on the average, 69.05 miles. 
Multiplying this by 360, we obtain 24,858 miles for the circum- 
ference of a meridian, and dividing this circumference by -, 
(3.1416) we find the length of the earth's diameter to be 7912 
miles. 

64. Spheroidal Form of the Earth. — One remarkable fact is 
noticed when we compare the lengths of the degrees of the meri- 
dian, measured in different latitudes; and that is, that the length 
of the degree is not the same at all parts of the meridian, but sen- 
sibly increases as we leave the equator. The length of a degree 
at the equator is found to be 68.7 miles, whilst at the poles it is 
computed to be 69.4 miles. The conclusion drawn from this 
fact is that the figure of the earth is not rigorously that of a 
sphere, since a spherical form necessarily implies an absolute 
uniformity in the length of a degree in all parts of a great circle. 
In order to determine the exact geometrical figure of the earth, 
we must bear in mind that the curvature of a line is always pro- 
portional to the change in the direction of the tangents drawn 
at successive points of that line. Now, since the altitude of the 
elevated pole at any place is equal to the latitude of that place, 
it follows that an advance towards the pole of one degree in lati- 
tude is accompanied by a depression of one degree in the plane of 
the horizon. If, therefore, in order to effect a change of one de- 
gree in our latitude, we are forced to advance a greater number 
of miles at the pole than at the equator, we conclude that the cur- 
vature of the meridian is less at the pole than at the equator. Now, 

this same inequality in its curvature is 
also a peculiarity of the ellipse : and hence 
we infer that the form of the earth's me- 
ridians is not that of a circle, but that 
of an ellipse, as represented in Fig. 26. 
The axis of the earth, Pp, corresponds 
to the minor axis of an ellipse, at the 
extremities of which the curvature is the 




DENSITY. 63 

least; and the equatorial diameter of the earth, EQ, corresponds 
to the major axis of an ellipse, at the extremities of which the 
curvature is the greatest. 

The form of the earth, then, is that of the solid which would 
be generated by the revolution of an ellipse about its minor 
axis, which solid is called in Geometry an oblate spheroid, A 
more common but less accurate name given to the form of the 
earth is that of a sphere, flattened at the poles. 

65. Dimensions of the Earth. — The following are the dimen- 
sions of the earth, when its spheroidal form is taken into con- 
sideration. The determination is that of Mr. Airy, the late 
Astronomer Royal of England. 

Polar diameter 7899.170 miles. 

Equatorial diameter 7925.648 miles. 

These values are believed to be within a quarter of a mile of the 
true values. They differ from the results obtained by the astro- 
nomer Bcss^l by only about ^th of a mile. 

The compression, or oblateness, of an oblate spheroid is the 
ratio of the difference between the major and the minor axis 
of the generating ellipse to its major axis. The compression of 

the earth is therefore -.~^' nAC , : which is about ;^:th. 
7925.648 299 

If a and b represent the semi-major and the semi-minor axis 

of the generating ellipse, the expression for the volume of the 

oblate spheroid is j~a 2 b. Substituting in this expression the 

values of a and b, we find the earth's volume to be about 260 

billions of cubic miles. 

66. Density of the Earth. — There are various snethods of de- 
termining the mean density of the earth. The following is a 
brief summary of the method of determining it by means of the 
torsion balance. This balance consists of a slender wooden rod, 
supported in a horizontal position by a very fine wire at its 
centre. To the extremities of this rod are attached two small 
leaden balls. If left free to move, this horizontal rod will of 
course come to rest when the supporting wire is free from 
torsion. Two much larger leaden balls are now brought near 
the two suspended balls, and on opposite sides, so that the 
attractions of both balls may combine to twist the wire in the 



64 ROTATION. 

same direction. The smaller balls will be sensibly attracted 
by the larger ones, and the horizontal rod will change the 
direction in which it lies. The amount of this deflection is 
very carefully measured, and from it is computed the attraction 
which the large balls exert on the small ones. But we know 
the attraction which the earth exerts on the small balls, it being 
represented by their weight: and we know also the volumes 
of the earth and the attracting balls. Finally, we know the 
density of lead: and from these data it is possible to compute 
the mean density of the earth. 

A series of over 2000 experiments of this nature was con- 
ducted in England, in 1842, by Sir Francis Baily. The mean 
density of the earth, obtained from these experiments, w 7 as 5.67 : 
the density of water being the unit. Other methods of deter- 
mining the density of the earth have been employed, the main 
principle in each method being the comparison of the at- 
traction exerted by the earth upon some object with that ex- 
erted by some other body, whose density can be ascertained, 
upon the same object. The results of these experiments do not 
differ materially from the results of the experiments with the 
torsion balance. 

The volume and density of the earth being known, what is 
commonly called its w 7 eight can be computed. It is found to 
be about six sextillions of tons. 

ROTATION OF THE EARTH. 

67. Up to this point w 7 e have assumed the earth to be at 
rest, and the apparent diurnal motions of the heavenly bodies 
to be real motions. By careful observation of the sun, the 
moon, and the most conspicuous of the planets, astronomers 
have demonstrated that each of these bodies rotates upon a 
fixed axis. Analogy, therefore, points to a similar rotation of 
our own planet: and besides this, there are many phenomena 
which are inexplicable if the earth is at rest, but which are 
fully accounted for on the supposition that it rotates upon an 
axis. We will now examine the principal of these phenomena. 

68. The weight of the same body is not the same in differeni 
latitudes. Careful experiments made in different latitudes show 



CENTRIFUGAL FORCE. 



65 



that the weight of the same body is not constant at all parts 
of the earth's surface, but increases with the latitude. A body 
which weighs 194 pounds at the equator will weigh 195 pounds 
if taken to either pole ; that is to say, the weight of any body is 
increased by y^th of itself when carried from the equator to 
the pole. This experiment cannot be made with the ordinary 
balances in which bodies are weighed: since it is obvious that 
the same cause, whatever it may be, which affects the weight 
of the body will also affect that of the weights by which it is 
balanced, and by the same amount, so that the scales will still 
remain in equilibrium. If, however, we test the weight of a 
body (the force, that is to say, ivith which it tends to the earth's 
centre) by the effect which it has in stretching a spring, the 
increase of weight will be found to be as stated above. 

Part of this increase of weight is due to the spheroidal form 
of the earth, since a body when at the pole is nearer the centre 
of the earth than when at the equator. The amount of increase 
due to this cause has been calculated to be about 5^o tn > nence 
the difference between y^^th and ^oth, which is ^-g-gth, still re- 
mains to be accounted for. We shall now see how it is com- 
pletely accounted for by the supposition that it is the effect of 
the centrifugal force which is induced by a rotation of the earth 
upon its polar axis. 

69. Centrifugal Force. — The tendency which a body has, when 
revolving about any point as a centre, to recede from that centre, 
is called its centrifugal force. The formula for the centrifugal 
force may be found in any treatise on Mechanics, and is as 
follows : 

/ = — 

in which / is the centrifugal force, r the 
radius of the circle of revolution, and t the 
periodic time, or the time in which the 
revolution is performed. Now, in Fig. 27 
let the earth be supposed to rotate about 
its polar axis, Pp, once in eve.ry sidereal 
day, which, as we have already seen 
(Art. 7), is 3m. 56s. loss than the mean 

5 




66 CENTRIFUGAL FORCE. 

solar day, and therefore contains 86,164s. Substituting this 
value of t in the formula given above, and substituting for r the 
value of the earth's equatorial radius in feet, and computing 
the value of/, we shall find that the centrifugal force at the 
equator is .1113 feet. Now the actual force of gravity at the 
equator is found, by Mechanics, to be 32.09 feet. If the earth 
were at rest, the force of gravity at the equator would evidently 
be 32.09 + .1113 feet. Hence the diminution of gravity at 
the equator, due to centrifugal force, (in other words, the loss 
of weight), is equal to s'^hii 3 > or sf 9 tn - 

Since the periodic time (t in the formula) is constant for all 
places on the earth's surface, it is evident, from the formula, that 
the centrifugal force at any place L is to the centrifugal force 
at the equator as the radius of revolution at L, or LM, is to CQ 
But we have in the figure, 

ML ML . - T . 1 

Yuj- = -j^r ~ eos = cos -L' atitu ^ e - 

Denying, then, the centrifugal force at the equator by C\ 
and that at L by c, we have, — 

c = C cos Tj : 
or the centrifugal force varies with the cosine of the latitude. 

The centrifugal force at L acts in the direction of the radius 
of revolution ML. Let its amount be represented by IjB, 
taken on LM prolonged. This force may be resolved into two 
forces : LA, in the direction from the centre of the earth, and A B, 
at right angles to LA. The force IjA, being directly opposed 
to the attraction of the earth, has the effect of diminishing the 
weight of bodies at L, and may therefore be taken to represent 
the loss of weight at L. 

Denoting the loss of weight by w, and the centrifugal force 
at L by c, as before, we have, from the triangle ABL y 

w = c cos L. 

But w T e have already, — 

c = C cos L : 
,-. iv = C cos' 2 L. 

Now, at the equator, as is evident from the figure, the whole 
effect of the centrifugal force is exerted to diminish the weight 
of bodies, and C therefore also represents the loss of weight at 



TRADE WINDS. 67 

the equator. We have then, finally, that the loss of weight of a 
body at any latitude-, due to centrifugal force, is equal to the pro- 
duct of oi^th of the weight multiplied by the square of the cosine 
of the latitude. 

70. Spheroidal Form of the Earth due to Centrifugal Force. — 
We see, then, that the supposition that the earth rotates upon its 
axis fully explains the observed difference in the weight of the 
same body in different latitudes. But this is not all : for if we 
assume that the particles of matter of which the earth is com- 
posed were formerly in a fluid or molten condition, and there- 
fore free to move, the spheroidal form of the earth is itself a 
proof of the earth's rotation. Numerous experiments may be 
made to show that, for a fluid body at rest, the form of equili- 
brium is that of a sphere : and that, for a fluid body which 
rotates, the form of equilibrium is that of a spheroid, the oblate- 
ness of which increases with the velocity of rotation. Knowing 
the volume and the density of the earth, and assuming the time 
of rotation to be twenty-four sidereal hours, it is possible to 
calculate the form of equilibrium which a fluid mass under 
these conditions will assume : and this form is found to be that 
of a spheroid, with an oblateness very nearly identical with the 
known oblateness of the earth. 

This tendency of a fluid mass to assume a spheroidal form 
under rotation may also be shown in Fig. 27. The centrifugal 
force LB was resolved into the two forces LA and AB, the 
former of which forces has already been discussed. The effect 
of the latter force, AB, is evidently a tendency in the particle 
L to move towards the equator EQ; and a similar force acting 
upon all the particles of matter on the earth's surface, excepting 
those at the poles and at the equator, will cause them all to 
move in the direction of the equator, and thus give a spheroidal 
form to the mass. 

71. Trade Winds. — The trade winds are permanent winds 
which prevail in and sometimes beyond the torrid zone. These 
winds are northeasterly in the northern hemisphere and south- 
easterly in the southern hem Sphere. The air within the torrid 
zone being, in general, subject to a greater degree of heat than 
the air at other portions of the earth's surface, rises, and its 



68 PENDULUM EXPERIMENT. 

place is filled by air which comes in from the regions beyond the 
tropics. If the earth were at rest, these currents of air would 
manifestly have simply a northerly and a southerly direction. 
Now, we all know that, when w 7 e travel in any direction on a 
still day, or even when the wind is moving in the same direc- 
tion with us, but with a less velocity, the wind seems to come 
from the point towards which we are going. We see from Fig. 
27 that, if the earth is rotating upon its polar axis, the linear 
velocity of rotation decreases as the latitude increases. Hence, 
the air from beyond the tropics, having at the start only the 
linear velocity of the place which it leaves, will, as it moves 
towards the equator, have continually a less velocity than that 
of the surface over which it passes, and will seem to come from 
the quarter towards which those places are moving. If, then, 
the earth is rotating from west to east, these currents of air will 
have an apparent motion from the east, w 7 hich motion, when 
compounded with the motion from the north and the south, 
before mentioned, will give us the northeasterly and south- 
easterly winds which we call the Trades. 

72. The Pendulum Experiment* — The last and decidedly the 
most satisfactory proof of the earth's rotation which we shall 
notice, is that w T hich comes from the apparent rotation of the 
plane of a freely-suspended pendulum, when made to vibrate 
at any point on the earth's surface except the equator. 

It is an established law in Mechanics that a pendulum, freely 
suspended from a fixed point, always vibrates in the same plane ; 
and also that if we give the point of support a slow movement 
of rotation about a vertical axis, the plane of vibration will still 
remain unchanged. If, for instance, w r e suspend a ball by a 
string, and, having caused it to vibrate, twist the string, the ball 
will rotate about the axis of the string, while the plane in which 
it vibrates will not be affected. 

Now, let us suppose that a pendulum is suspended at the north 
pole, and is made to vibrate: and let us further suppose that the 
earth rotates from west to east, once in 24 hours. The line in 
which the plane of vibration intersects the plane of the horizon 

* This is called FoucauWs experiment. A full discussion of it is given in 
the American Journal of Science, 2d series, vols. XII-XIV. 




PENDULUM EXPERIMENT. 69 

Will move about in the plane of the horizon, in a direction oppo- 
site to that in which the earth is rotating, and with an equal 
velocity, thus completing one revolution in 24 hours. In Fig. 
28, let ACBD be the horizon of the ob- 
server at the north pole, and let the earth 
rotate in the direction indicated by the 
arrows. Let the pendulum at P be set 
swinging in the direction of some diameter, 
AB, of the horizon. At the end of an hour, 
the rotation of the earth will have carried 
this diameter to some new position A B\ 
at the end of the next hour to some new position A"B", &c. : 
while the pendulum will still swing in the original direction AB. 
To the observer, then, unconscious of the earth's rotation, the 
plane of vibration, which really remains unchanged, will appear 
to rotate in a direction opposite to that in which the earth is 
rotating. 

At the south pole, under the same suppositions, a similar phe- 
nomenon will be noticed, except that the plane of vibration will 
apparently move in the opposite direction. Thus, if at the north 
pole the apparent motion of the plane is like that of the hands 
of a clock, as we look on its face, the apparent motion at the 
south pole will be the opposite to this. 

Again, if a pendulum is made to vibrate in the plane of a 
meridian at the equator, there will be no apparent change in the 
plane of vibration, since it will always coincide with the plane 
of the meridian, and hence the pendulum will continue to 
swing north and south during the entire period of the earth's 
rotation. The condition that the pendulum shall here swing in 
the plane of a meridian is entirely unnecessary, and is made 
only for the sake of illustration; for there will be no apparent 
change in the plane of vibration, whatever may be the direction 
in which the pendulum is made to vibrate. 

The apparent rotation, then, of the plane of vibration of the 
pendulum is 360° in 24 hours at the poles, and nothing at the 
equator. At places lying between the equator and the poles, 
the apparent angular motion of the plane of vibration will be 
between these two limits; in other words, less than 360° in 



70 LINEAR VELOCITY OF ROTATION. 

24 hours. Appropriate investigations show that the apparent 
angular motion of the plane of vibration at any place in any 
interval of time is equal to the angular amount of the earth's 
rotation in that time, multiplied by the sine of the latitude of 
the place.* Thus, at Annapolis, we have for the angular motion 
in one hour, 

15° sin 38° 59' = 9° 26': 
so that the plane of vibration will make one apparent rotation 
at Annapolis in 38h. 09m. 

Such is the theory of the pendulum experiment. Now, nume- 
rous experiments have been made in different latitudes, and in 
every case an apparent rotation of the plane of vibration from 
east to west has been observed, with a rate agreeing very closely 
with that demanded by the theory ; and the conclusion is irre- 
sistible that the earth rotates on its polar axis, from west to east, 
once in every sidereal day. 

73. Linear Velocity of Rotation. — Taking the equatorial cir- 
cumference of the earth to be 24,900 miles, we have a linear 
velocity of over 1000 miles an hour, and over 17 miles a minute. 
This is the velocity at the equator. The linear velocity at other 
points on the earth's surface is less than this, since the circum- 
ferences of the parallels of latitude are less than the circumfer- 
ence of the equator. Since the circumference of any parallel is 




* This formula may be obtained by the principles of the resolution of ro- 

tation, given in treatises on Mechanics. Thus, in the 

figure, the rotation of the point L about the axis of 

the earth, PO, may be resolved into two rotations, 

one about the radius LO, and the other about the 

radius MO, drawn perpendicular to LO. If v re- 

^ presents the angular velocity of L about the axis 

PO (or 15° in one hour), and v / and v" the angular velocities about the 

axes LO and MO, we have, from Mechanics, 

v' = v cos LOP, and v" -= v cos POM. 
Now, the rotation about the axis OM will have no effect in changing the 
apparent position of the plane of vibration of the pendulum, since it is 
analogous to the case at the equator considered in the text; while the rota- 
tion about the axis LO, being analogous to the case at the pole, will pro- 
duce a similar effect. The apparent angular motion, then, of the plane 
of vibration will be v cos LOP, or v sin Lat. 



LINEAR VELOCITY OF ROTATION. 71 

to that of the equator as the radius of the parallel is to the 
radius of the equator, the linear velocity will diminish as we 
leave the equator in the same ratio that the radii of the succes- 
sive parallels diminish : in the ratio, that is, of the cosine of the 
latitude, as was proved in Art. 69. For instance, the cosine of 
60° being £, the linear velocity at that latitude is only 8 J miles 
a minute. 



LATITUDE. 



CHAPTER V. 

LATITUDE. LONGITUDE. 



LATITUDE. 

74. The latitude of any place on the earth's surface has been 
proved, in Articles 10 and 11, to be equal to either the altitude 
of the elevated pole or the declination of the zenith at that 
place. We shall now proceed to explain the principal methods 
by which either one or the other of these arcs may be found. 

75. First Method. — Let Fig. 29 represent a projection of the 
celestial sphere on the plane of the celestial 
meridian, RZHN, of some place. HE is 
the celestial horizon at that place, Z the 
zenith, P the elevated pole, and EQ the 
equator. Let s represent some circumpolar 
star, whose declination is known, at its 
lower culmination. Let its meridian alti- 
tude be observed, and corrected for instru- 
mental errors and refraction. (For all celestial bodies except the 
sun, the moon, and the planets, the corrections for parallax and 
semi-diameter will be inappreciable.) To this corrected altitude 
add the star's polar distance, the complement of the star's known 
declination. The sum is the altitude of the elevated pole, or 
the latitude. 

If the circumpolar star is at its upper culmination, as at s\ 
the polar distance is to be subtracted from the corrected altitude. 
If h! and h denote the corrected altitudes at the upper and 
the lower culmination, p f and p the corresponding polar dis- 
tances, and L the latitude, we have evidently 
L = h' —p' 
L = h+p: 
whence L = i (h f + K) + 1 (p — p). 




LATITUDE. 73 

In this formula the value of the latitude does not depend on the 
absolute value of either polar distance, but merely on the change 
of the polar distance between the two transits, which is usually 
so small as to be neglected. This method, then, is free from any 
error in the declination, and is used at all fixed observatories. 

76. Second Method. — When the star is at its upper culmina- 
tion, it will, in general, be more convenient to find the declina- 
tion of the zenith from the meridian zenith distance of the star. 
Taking the star s' , for instance, and denoting its meridian zenith 
distance by z, and its declination by d, we have 

L = ZQ= Qs f — Zs' = d — z. (a) 

For the star s", we have 

L = Zs" + Qs" =z + d, (6) 

and for the star s" 

L = Zs'" — Qs'" = z — d. (c) 

From these three formulae a general rule may be deduced, appli- 
cable to the upper culmination of every star. We notice that 
in the formulae (a) and (6), where d is positive, the stars s and 
s" are on the same side of the equator with the elevated pole; 
that is to say, their declinations have the same name as the ele- 
vated pole; while in the formula (c) the declination has the 
opposite name. We also notice that in the formulae (6) and (c), 
where z is positive, the stars are on the opposite side of the zenith 
from the elevated pole; in other words, their bearing has the op- 
posite name to that of the pole : while the bearing of the star s', 
in the formula for which z is negative, has the same name as the 
elevated pole. The general rule, then, for all these stars will be 
the following: — If the star bears south, mark the zenith distance 
north; if it bears north, mark the zenith distance south; mark 
the declination north or south, as the star is north or south of 
the equator, and combine the zenith distance and the declina- 
tion, thus marked, according to their names. 

77. Third Method. — A very successful adaptation of the pre- 
ceding method is made by using two stars which culminate at 
nearly the same time, but on opposite sides of the zenith, as s 9 
and s" in Fig. 29. These two stars are so selected that the dif- 
ference of their zenith distances is very small, and can be mea* 



74 LATITUDE. 

sured directly by means of a micrometer. By the formulae of 
the preceding article we have for s', 

L = d — z, 
and for $'\ denoting its meridian zenith distance and declination 
by z and d\ 

L = d' + z\ 
whence we have, 

L = l(d+ d') + i(z — *). 
The determination of the latitude is thus made free from any 
error in the graduations of the vertical circle, and depends only 
on the known declinations of the two stars, and on the difference 
of their zenith distances. Errors in the refraction are also very 
nearly eliminated. 

This is the principle of what is called Talcott's Method, a 
method very commonly used by the United States Coast Survey. 
The instrument employed is the zenith telescope, a modification 
of the altitude and azimuth instrument. The two stars are so 
selected that the difference of their zenith distances is less than 
the breadth of the field of the telescope. The instrument is set 
in the plane of the meridian to the mean of the two zenith dis- 
tances, and for the star which culminates first. When this star 
crosses the meridian, it is bisected by the micrometer wire, and 
the micrometer is read. The instrument is then turned 180° in 
azimuth, and the process is repeated with the second star. The 
difference of the zenith distances is then obtained from the dif- 
ference of the two micrometer readings, and added to the half 
sum of the two declinations, according to the formula. 

78. Fourth Method. — When the local time (either solar or 
sidereal) is known, the latitude may be obtained from altitudes 
which are not measured on the meridian. Let Fig. 30 be a pro- 
jection of the celestial sphere on the plane 
of the horizon. Z is the zenith of the place, 
P the elevated pole, PZ the co-latitude, and 
S a star, whose altitude is measured. SPZ 
is the hour angle of the star, which can be 
obtained from the local time noted at the 
instant the altitude is observed. PS is the 
star's known polar distance. In the triangle 




REDUCTION OF THE LATITUDE. 



10 



SPZ, we have the sides ZS and SP, and the angle SPZ, and can 
therefore compute the value of the co-latitude, PZ, by the for- 
mulae of Spherical Trigonometry. 

An analytical investigation of the formula? by which this pro- 
blem is solved shows that errors in the observed altitude and the 
time have the less effect upon the result the nearer the body is 
to the meridian. 

79. Methods of Finding the Latitude at Sea. — The second and 
the fourth of the methods above described are the methods most 
commonly employed in finding the latitude at sea. The sun \< 
the body which is generally used, its altitude above the sea 
horizon being measured with a sextant or an octant. The time of 
noon being approximately known, the observer begins to measure 
the altitude of the lower limb of the sun a few minutes before 
noon, and continues to measure it until the sun ceases to rise, or 
"dips," as it is called. The greatest altitude which the sun 
attains is considered to be the meridian altitude, although, rigor- 
ously speaking, it is not. The proper corrections for index-error, 
dip, refraction, parallax, and semi-diameter are next applied to 
the sextant reading, and the result is the sun's true meridian 
altitude, from which the latitude is obtained by the rule given in 
Art. 76. 

When cloudy w T eather prevents the determination of the meri- 
dian altitude of either the sun or any other celestial body, an 
altitude obtained w r ithin an hour of transit, on either side of the 
meridian, maybe used to find the latitude by the fourth method. 
Art. 78. Bowditch's Navigator contains special tables by which 
the computation, particularly when the sun is observed, may be 
greatly facilitated. 

80. Reduction of the Latitude. — 
Owing to the spheroidal form of the 
earth, the vertical line at any point 
of the surface, as Z' O in Fig. 31, 
which corresponds exactly with 
the normal drawn at that point, 
does not coincide with the radius of 
the earth, LO, passing through the 
same point, excepting at the equator 




<0 LONGITUDE. 

and the poles. It is necessary, then, in refined observations, to 
distinguish between the geographical zenith, Z', the point where 
the vertical line, when prolonged, meets the celestial sphere, and 
the geocentric zenith, Z, the point in which the radius meets the 
sphere. Since there are two zeniths, there are also two lati- 
tudes : Z' 0' Q, the geographical latitude, and ZOQ, the geocentric 
latitude. The geographical latitude is evidently greater than 
the geocentric, by the angle OLO\ which is called the reduction 
of the latitude. Formulae and tables for finding this reduction 
are given in Chauvenet's Astronomy. It is only considered in 
cases where the highest accuracy in the results is required. 



LONGITUDE. 

81. Let Fig. 32 represent a projection of the celestial sphere 
on the plane of the equinoctial ABCG. P is the projection of 
the elevated pole, and PG, PA, and PB are 
projections of arcs of great circles of the 
sphere passing through the pole. Let PG 
represent the projection of the meridian of 
Greenwich, PA that of the meridian of some 
other place on the earth's surface, and PB 
that of the circle of declination passing 
through some celestial body S. Then will 
the angle GPA represent the longitude of the meridian PA 
from Greenwich, GPB will represent the Greenwich hour angle 
of the body S, and APB will represent its hour angle from the 
meridian PA. The difference between these two hour angles is 
evidently equal to the longitude of any place on the meridian 
PA. The longitude, then, of any place on the earth's surface 
is equal to the difference of the hour angles of the same celestial 
body at that place and at Greemvich, at the same absolute instant 
of time. When the Greenwich hour angle is the greater of these 
two hour angles, reckoned always to the west, the longitude of 
the place is west : when it is the smaller, the longitude is east. 

If PB is the hour circle passing through the sun, the longi- 
tude of the place is the difference of the solar times at the place 
and at Greenwich : if it is the hour circle passing through the 




CHRONOMETERS. 7 i 

vernal equinox, the longitude is the difference of the two side- 
real times. In order, then, to determine the longitude of any 
place, we must be able to determine both the local and the 
Greenwich time (either sidereal or solar) at the same instant. 

There are various methods of obtaining the local time, one 
of which has already been described (Art. 20). It may be 
noticed here that w T e are always able, by means of the Nautical 
Almanac, to convert sidereal time into solar time, or solar into 
sidereal (Art. 105). It remains, then, to determine the Green- 
wich time, either sidereal or solar, to do which several distinct 
methods may be employed. 

82. Greenwich Time by Chronometers. — If a chronometer is 
accurately regulated to Greenwich time, that is to say, if the 
amount by which it is fast or slow at Greenwich on any day, 
and its daily gain or loss, are determined by observation, the 
chronometer can be carried to any other place the longitude 
of which is desired, and the Greenwich time which the chrono- 
meter gives can be directly compared with the time at that place. 
This would be a perfectly accurate method, if the rate of the 
chronometer remained constant during the transportation; but, 
in fact, the rate of a chronometer while it is carried from place 
to place is very rarely exactly the same that it is w r hile the 
chronometer is at rest. By using several chronometers, however, 
and by transporting them several times in both directions be- 
tween the two places, and finally by taking a mean of all the 
results, the error may be reduced to a very minute amount. 
For instance, the longitude of Cambridge, Mass., was deter- 
mined by means of fifty chronometers, which were carried three 
times to Liverpool and back, and from them the longitude was 
obtained with a probable error of only 4th of a second of time. 

83. Greenwich Time by Celestial Phenomena. — There are cer^ 
tain celestial phenomena which are visible at the same absolute 
instant of time, at all places where they can be seen at all. Such 
are the beginning and the end of a lunar eclipse ; the eclipses 
of the satellites of the planet Jupiter by that planet; the 
transits of these satellites across the planet's disc, and their oc- 
cupations by it. The Greenwich times at which these various 
phenomena will occur are computed beforehand, and are pub- 



78 LUNAR DISTANCES. 

lished in the Nautical Almanac. The observer, then, to obtain 
his longitude, has only to note the local time at which any one 
of these phenomena occurs, and to compare that time with the 
corresponding Greenwich time given in the Almanac. The dif- 
ficulty of determining the exact instant at which these phe- 
nomena occur, however, diminishes to some extent the accuracy 
of the results. On the other hand, the times of solar eclipses 
and of occupations of stars by the moon, although not identical 
at different places, can be very accurately determined : and 
hence these phenomena are often employed in obtaining longi- 
tudes. (Art. 164.) 

84. Greenwich Tims by Lunar Distances. — By the lunar distance 
of a celestial body is meant its true angular distance from the 
centre of the moon, as it would be seen at the centre of the 
earth. The lunar distances of the sun, of the four brightest 
planets, and of nine bright stars are given in the Nautical 
Almanac, computed for every third hour of Greenwich solar 
time. An observer, then, who wishes to determine his longitude, 
measures the apparent angular distance of the moon from some 
one of these bodies, and also notes the local time at which the 
observation is made. He then finds from this apparent distance, 
by means of appropriate formulae and tables, the true geocentric 
angular distance, at the time of observation, between the two 
bodies. He then enters the table of lunar distances in the 
Almanac with this distance, and finds the corresponding Green- 
wich time, from which, and the local time noted, he can deter- 
mine his longitude. 

85. Difference of Longitude by Electric Telegraph. — When two 
stations, the difference of longitude of which is desired, are 
connected by an electro-telegraphic wire, the difference of longi- 
tude may be determined by means of signals made at either 
station, and recorded at both. Suppose, for instance, there are 
two stations A and B, of which A is the more easterly, and 
that each station is provided with a clock regulated to its own 
local time. Let the observer at A make a signal, the time of 
which is recorded at each station. Let A denote the difference 
of longitude of the two stations, T the local time at A at which 
the signal is made, and T' the corresponding time at B. Since 



ELECTRIC SIGNALS. 79 

A is to the east of B, its time is greater at any instant than 
that of B. We have then, supposing the signal to be recorded 
simultaneously at the two stations, 

x = t— r. 

Experience proves, however, that the records of the signal 
are not exactly simultaneous, since time is required for the 
electric current to pass over the wire. In the example above 
given, then, if we denote the time required by the electric fluid 
to pass from A to B by x, the time recorded at B will evidently 
be, not T', but T' + x ; so that the expression for the difference 
of longitude will be 

X = T—T'—x. 

Now let us suppose that instead of the signal's being made 
by the observer at A, it is made by the observer at B, at the 
time T' . The corresponding time recorded at A will not be T, 
but T+ x. In this case, then, the expression for the difference 
of longitude will be 

;/' = t+ x— r. 

Taking the mean of the values of X' and /", we have 
i (;/ + ;/') = T— T'=L 

Any error, therefore, which is caused by the time consumed 
by the electric current in passing between two stations is elimi- 
nated by determining the difference of longitude by signals made 
at both stations, and taking the mean of the results. 

86. Difference of Longitude by "Star Signals." — The " method 
of star signals" is a modification of the method described in 
the preceding paragraph, which is extensively used in the 
United States Coast Survey. The principle on which this 
method rests is that, since a fixed star makes one apparent revo- 
lution about the earth in exactly twenty-four sidereal hours, 
the difference of longitude between two meridians is equal to 
the interval of sidereal time in which any fixed star passes from 
one of these meridians to the other. The clock by which this 
interval of time is measured may be placed at either station, 
or indeed at any place which is in telegraphic communication 
with both stations. Two chronographs, one at each station, are 
connected with the wire and the clock, and upon them are 



80 LONGITUDE AT SEA. 

recorded, by breaks in the circuit as explained in Art. 22, the 
successive beats of the clock. A transit instrument is adjusted 
to the meridian at each station. As the star crosses the several 
threads of the reticule of the transit instrument at the eastern 
station, the observer, by means of a break-circuit key, records 
the instants upon both chronographs. The same process is 
repeated as the star crosses the wires at the western station. 
Now, it is evident that the elapsed time between the transits at 
the two meridians has been recorded upon each chronograph. 
Each of these values of the elapsed time is to be corrected for 
instrumental errors, errors of observation, and for the gain or 
loss of the clock in the interval; and the mean of the two 
values, thus corrected, is taken as the difference of longitude of 
the tw r o places. 

By making similar observations on several stars on the same 
night, by repeating the observations on subsequent nights, by 
exchanging observers and using different clocks, and, finally, 
by taking a mean of the results, a very accurate determination 
of the difference of longitude may be secured. 

87. Method of Finding the Longitude at Sea. — The method of 
finding the longitude at sea which is usually employed is the 
method of Art. 82. The Greenwich time is given by chro- 
nometers regulated to Greenwich time, and the local time is 
obtained from the observed altitudes of celestial bodies. The 
sun is the body the altitude of which is most commonly used 
for this purpose; but altitudes of the most conspicuous of the 
planets and the fixed stars may also be successfully employed. 
Altitudes of the moon are to be avoided, except in cases where 
no other body is available. At the instant when the altitude 
of any celestial body is observed, the time shown by a watch is 
noted. This watch, either shortly before or after the observa- 
tion, is compared with the Greenwich chronometer, and by 
means of this comparison the Greenwich time of the observa- 
tion is obtained from the time given by the watch. The neces- 
sary corrections are applied to the sextant reading to obtain 
the body's true altitude. We shall then have, in the triangle 
PZS, Fig. 33, the side ZS, the zenith distance of the body, PS 
its polar distance, obtained from the Nautical Almanac, and 




COMPARISON OF LOCAL TIMES. 81 

PZthe co-latitude of the place of observation ; 

the latitude being determined by some one of 

the methods already given (Art. 79), and being 

reduced to the time of observation by the run d\ 

of the ship given by the log. In the triangle 

PZS, then, having the three sides given, we 

can compute the angle SPZ, which is the hour 

angle of the body. From this hour angle the 

local time can be readily found, from which, and the Greenwich 

time already obtained, the longitude may be determined. 

In case there is no chronometer on board, the method of 
lunar distances is the only regularly available method of deter- 
mining the Greenwich time. At the present day, how T ever, 
lunar distances are mainly employed as checks upon the chro- 
nometer, since any change in the rate of a chronometer will 
cause a discrepancy between the Greenwich time shown by the 
chronometer and that deduced from observation. 

It can be shown, by proper methods of investigation, that 
an error in the assumed latitude, or in the body's altitude, 
causes the less error in the resulting hour angle the nearer 
the body is to the prime vertical. It is best, then, in observing 
the altitude of any celestial body for the purpose of obtaining 
the local time, to observe it when the body bears nearly east or 
west, provided the altitude is not so small as to be sensibly 
affected by errors in the refraction. It may also be shown that 
in selecting celestial bodies for observations of this character, it 
is best, if the other conditions are satisfied, to take those bodies 
which have the smallest declinations. 

88. Comparison of the Local Times of Different Meridians. — 
Since the local time, either solar or sidereal, is the greater 
at the more easterly of any two meridians, it follows that a 
watch or chronometer which is regulated to the time of any 
one meridian will appear to gain when carried to the west, 
and to lose when carried to the east : the amount of gain or 
loss in any case being the difference of longitude, in time, 
of the two meridians. A watch, for instance, which gives the 
correct solar time at Boston will, even if it really is running 
accurately, appear to gain nearly twelve minutes when taken 



82 COMPARISON OF LOCAL TIMES. 

to New York. If, then, a watch which is regulated to the solar 
time of any meridian is carried to the east, the difference of 
longitude in time between the meridian left and that arrived 
at must be added to the reading of the watch, to obtain the 
time at the second meridian: if it is carried to the ivest, the 
difference of longitude must be subtracted. 



ECLIPTIC. 



83 



CHAPTER VI. 



THE SUN. THE EARTHS ORBIT. THE SEASONS. 
THE ZODIACAL LIGHT. 



TWILIGHT. 



89. The Ecliptic. — If a great circle on any globe is assumed 
to represent the celestial equator, and any point of that circle 
is taken to represent the vernal equinox, the relative positions 
of all bodies, the right ascension and declination of which are 
known, can be plotted upon this globe, and we shall have a 
representation of the celestial sphere. The poles of the great 
circle will represent the poles of the celestial sphere, and all 
great circles passing through these poles will represent circles 
of declination. We have seen, in the chapter on Astronomical 
Instruments, in what manner the right ascension and the decli- 
nation of any celestial body can be determined at any time by 
observation. If we thus determine the position of the sun from 
day to day, and mark the corresponding points upon our celes- 
tial globe, we shall find that the sun appears to move in a great 
circle of the sphere from west to east, completing one revolution 
in this circle in 365d. 6h. 9m. 9.6s. of our ordinary solar time. 
This interval of time is called the sidereal year. The great circle 
in which the sun appears to move is called the ecliptic, and the 
two points in which it intersects the celestial equator are called 
the vernal and the autumnal equinox. 

Let Fig. 34 be a representation of 
the celestial sphere. EA Q Fis the equi- 
noctial, Pp is the axis of the sphere, and 
P the north pole. The circle A CVD re- 
presents the ecliptic, V the vernal, and 
A the autumnal equinox. The sun is 
at the vernal equinox on the 21st of 
March, It thence moves eastward and 
northward, and reaches the point C, 




84 DISTANCE OF THE SUN. 

where it has its greatest northern declination, on the 21st of June. 
This point is called the northern summer solstice. From this point 
it moves eastward and southward, passes the autumnal equinox 
A on the 21st of September, and reaches the point D, called the 
northern winter solstice, on the 21st of December. It thence 
moves towards V, which it reaches on the 21st of March. 

The obliquity of the ecliptic to the equinoctial is the angle 
CVQ, measured by the arc CQ. This angle or arc is evidently 
equal to the greatest declination, either north or south, which 
the sun attains, and is found by observation to be about 23° 27'. 

90. Definitions. — The latitude of a celestial body is its angular 
distance from the plane of the ecliptic, measured on a great circle 
passing through its poles, and called a circle of latitude. In Fig. 
34 the arc Ks is the latitude of the body s. The longitude of a 
celestial body is the arc of the ecliptic intercepted between the 
vernal equinox and the circle of latitude passing through the 
body. Thus VK is the longitude of the body s. Longitude is 
properly reckoned towards the east. 

The hour circle which passes through the solstices, the circle 
DHCB, is called the solstitial colure. The hour circle which 
passes through the equinoxes is called the equinoctial colure. 

91. Signs. — The ecliptic is divided into twelve equal parts, 
called signs, which begin at the vernal equinox, and are named 
eastward in the following order: Aries, Taurus, Gemini, Cancer, 
Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, 
Pisces. Hence the vernal equinox is called the first point of 
Aries. 

The Zodiac is a zone or belt on the celestial sphere, extend- 
ing about 9° on each side of the ecliptic. 

DISTANCE OF THE SUN FROM THE EARTH. 

92. Relative Distances of the Earth and Venus from the Sun.— 
It is found by observation that the mean value of the sun's 
angular semi-diameter remains constant from year to year, 
being always 16' 2". Since any increase or decrease in the 
distance of the earth from the sun will evidently be accom- 
panied by a corresponding decrease or increase in the sun's 
angular semi-diameter, we conclude that the mean distance of 



DISTANCE OF THE SUN. 



85 



earth 



the earth from the sun is also constant from year to year. The 
distance of the earth from the sun is obtained by determining 
the sun's horizontal parallax from certain observations made 
upon the planet Venus. This planet revolves in a nearly cir- 
cular orbit about the sun, in a plane only 3° inclined to the 
plane of the ecliptic. Its distance from the sun is less than 
that of the earth from the sun, and hence it sometimes passes 
between the earth and the sun, and is seen apparently moving 
across the sun's disc. This phenomenon is called a transit of 
Venus. As a preliminary to the determination of the earth's 
distance from the sun from one of these transits, it is neces- 
sary to obtain the relative distances of Venus and the 
from the sun. To do this, in Fig. 35 let S 
be the sun, E the earth, and VV'V'V"' the 
orbit of Venus about the sun. It is evident 
that the greatest angular distance (or elon- 
gation) of Venus from the sun, the greatest 
value, that is, of the angle VES, will occur 
when the line from the earth to Venus is 
tangent to the orbit of Venus, as repre- 
sented in the figure. The orbit of Venus is 
not really a circle, but an ellipse, and hence 
the distance VS is slightly variable. So, 
also, is the distance SE; hence the greatest 
elongation is also variable, being found to lie between the limits 
of about 45° and 47°. Assuming its mean value to be 46°, we 
have in the right-angled triangle VSE, 

VS = SE sin 46°= .72 SE. 
Neglecting the inclination of the orbit of Venus to the plane 
of the ecliptic, we shall hare, at the time of a transit, when 
Venus is at V , 

V'E = .28 SE. 

Hence at the time of a transit the distance of Venus from the 
sun is to that of Venus from the earth as about 72 to 28.* 




* If we know the periodic time of Venus and that of the earth, the ratio 
of the distances of these two planets from the sun can be obtained by Kepler's 
Third Law (Art. 117), that "the squares of the periodic times of any two 
planets are proportional to the cubes of their mean distances from the sun." 



86 DISTANCE OF THE SUN. 

93. Transit of Venus. — In Fig. 36, let S denote the centre of 
the sun, and CADN its disc : let V be Venus, and E the centre 




Fig. 36. 

of the earth. Let HK be that diameter of the earth which is 
perpendicular to the plane of the ecliptic, and let an observer be 
supposed to be stationed at each extremity. In order to simplify 
the explanation, let us neglect the rotation of the earth during 
the observation, and suppose Venus to move in the plane of the 
ecliptic. To the observer at H, Venus will appear to move 
across the sun's disc in the chord CD, and to the observer at K, 
in the chord AB. Kegarding VHK and VFG as similar tri- 
angles, we have, by Geometry, 

FG : HK= GV: VH = 72 : 28 

18 
.-.FG = -^-HK. 

Again, w^e can obtain the angle which the line FG subtends 
at the earth's centre in the following manner. Let the observer 
at H note the interval of time in which the planet crosses the 
sun's disc in the chord CD, and the observer at K the interval 
in w r hich it moves through the line AB. Since there are tables 
which give us the angular velocity, as seen from the earth, both 
of the sun and of Venus, we can deduce the angles at the earth's 
centre subtended by the chords FB and GD, and knowing also 
the angular semi-diameter of the sun, in other words, the angle 
at the earth's centre subtended by SB or SD, we can compute 
the angles at the earth's centre subtended by FS and GS, and, 
finally, the angle subtended by FG. 

We have now determined the angle subtended by the line FG, 
at a distance equal to that of the earth from the sun, and also 
the ratio of FG to the earth's diameter. It is evidently easy to 
obtain from these values the angle at the sun subtended by the 



MAGNITUDE OF THE SUN. 87 

earth's radius, which angle is the sun's horizontal parallax, as 
we have already seen in Art. 54. 

Although we have assumed in this discussion that the two ob- 
servers are stationed at the extremities of the same diameter, it 
is really only necessary that they shall be at two places whose 
difference of latitude is large. The earth's rotation and other 
things which we have here neglected must be taken into con- 
sideration in the practical determination of the sun's parallax. 

94. Distance of the Earth from the Sun. — The last transit of 
Venus occurred in 1769, and from observations then made the . 
sun's horizontal parallax was determined to be 8".6. Later 
observations of a different character have given a horizontal 
parallax a little larger than this, 8". 848:* and this value is be- 
lieved to be very nearly exact. 

From Art. 54, we have for the distance in miles of the earth 
from the sun, 

d = R cosec P = 3962.8 cosec 8".848 = 92,400,000 miles. 

95. Magnitude of the Sun. — 
The length of the sun's radius 
can be at once obtained as soon 
as we know its distance from 
the earth. Thus, in Fig. 37, 
let S be the centre of the sun, Fi &- 37 - 

and E that of the earth. The angle AES is the apparent semi- 
diameter of the sun, which we obtain by observation, its mean 
value being, as already stated, 16' 2". We have then, in the 
right-angled triangle AES, 

SA = 92,400,000 sin 16' 2" = 431,000 miles. 
The sun's linear radius, then, is equal to nearly 109 of the earth's 
radii; and since the volumes of spheres are proportional to the 
cubes of their radii, the volume of the sun bears to that of the 
earth the enormous ratio of 1,286,000 to 1. 

By observations and calculations which will be described in 
the Chapter on Gravitation (see Art. 114), the mass of the sun is 
found to be about 327,000 times that of the earth ; or about 670 
times the sum of the masses of all the planets of the solar system. 

* Determination of Professor Simon Newcomb, United States Navy. The 
value obtained by Le Terrier is 8 /7 .95. 




88 



ORBIT OF THE EARTH. 



THE EARTHS ORBIT. 

96. Revolution of the Earth about the Sun. — Up to this point 
we have spoken of the apparent annual motion of the sun in the 
ecliptic from west to east, as though the earth were really at rest, 
and the sun revolved about it in its orbit. But when we take 
into consideration the immense mass of the sun compared with 
that of the earth, we are almost irresistibly led to conclude that 
the apparent annual revolution of the sun is the result, not of the 
actual revolution of the sun about the earth, but of tha.t of the 
earth about the sun. Such a revolution of the earth, from west 
to east,* would give to the sun precisely that apparent motion 

in the ecliptic which has been ob- 
served. This may be seen in Fig. 
38. Let S be the sun, EE'E" the 
earth's orbit, and the outer circle 
S'S ff S'" the great circle in which 
the plane of the ecliptic, indefi- 
nitely extended, meets the celestial 
sphere. When the earth is at E, 
the sun will be projected in 8'; 
when the earth is at E\ the sun 
will be projected in $", &c. ; that 
Fi s- 38 - is to say, while the earth moves 




* Whatever the absolute motion of any celestial body moving in a circle 
or an ellipse may be, the appearance presented in that motion will be re- 
versed if the spectator moves from one side of the plane in which the 
motion is performed to the other. Thus, the apparent daily motion to the 
westward of any celestial body is the same as the motion of the hands of a 
clock as we look upon its face, to an observer who is on the north side of 
the plane of the diurnal circle in which the body moves, as is seen in any 
latitude north of 23° 27' in the motion of the sun ; while the same west- 
ward motion presents the opposite appearance if the observer is to the south 
of the plane of motion, as may be seen in these latitudes in the case of the 
Great Bear. The appearance presented by a motion from west to east is 
of course the reverse of this ; hence when we say that the earth or any 
other body moves about the sun from west to east, we mean that, to an ob- 
server situated to the north of the plane of motion, the body appears to move in a 
direction opposite to that in which the hands of a clock move. 



OREIT OF THE EARTH. 89 

about the sun in the direction EE'E", the sun will apparently 
move about the earth in the same direction, S'S"S"'. 

Against this theory, then, of the earth's revolution there is 
nothing to urge; and analogy gives us a strong argument in 
favor of it. Almost every celestial body in which any motion 
at all can be detected is found to be revolving about some other 
body, larger than itself. The moon revolves about the earth; 
the satellites of the planets revolve about the planets; and the 
planets themselves, some of which are much larger than the 
earth, and at a much greater distance from the sun, revolve 
about the sun. Henceforward, then, we shall include the earth 
in the list of planets, and consider the sidereal year to be the 
interval of time in which the earth makes one complete revolu- 
tion about the sun. 

97. Linear Velocity of the Earth in its Orbit. — The number of 
miles in the circumference of the earth's orbit, considered as a 
circle, is obtained by multiplying the radius of the orbit by 2*. 
If w T e then divide this product by the number of seconds in a 
year, we shall have, in the quotient, the number of miles through 
which the earth moves about the sun in a second of time. It 
will be found to be about 18.4 miles. 

98. Elliptical Form of the Earth's Orbit. — Although, as has 
already been stated, the mean value of the sun's angular semi- 
diameter remains constant from year to year, careful measure- 
ments of the semi-diameter show that it varies in magnitude 
during the year, being greatest about the first of January, and 
least about the first of July. The evident conclusion from this 
fact is that the distance between the earth and the sun also varies 
during the year, being greatest when the sun's semi-diameter is 
the least, and least when it is the greatest. The truth of this 

-inclusion may be seen in Fig. 37, in which we have 

sin AES = jro- 

As AS of course remains constant, ES will vary inversely as sin 
AES, or since the sines of small angles are proportional to 
the angles themselves, inversely as the angle AES itself. The 
greatest angular semi-diameter of the sun is 16' 17". 8, the least 




90 ORBIT OF THE EARTH. 

is 15' 45". 5: hence the ratio of the greatest to the least distance 
is that of 16' 17".8 to 15' 45".5, or of 1.034 to 1. 

Let us now assume any line, SA in Fig. 39, for instance, as our 
unit of measure, and prolong it until 
SH is to SA as 1.034 is to 1. Then if S 
denotes the sun, SA and SH will repre- 
\ H sent the relative distances of the earth 
from the sun on about the first of Janu- 
ary and the first of July. On certain 
' D days throughout the year, let the advance 

Fig. 39. of the sun in longitude since the time 

when the earth was at A be determined, and let the angular 
semi-diameter of the sun on each of these days be measured. 
Lay off the angles ASB, ASC, &c, equal to these advances in 
longitude. Since, as may readily be seen in Fig. 38, the appa- 
rent advance of the sun in longitude is caused by the advance 
of the earth in its orbit, and is equal to it, the angles ASB, ASC, 
&c, w T ill represent the angular distances of the earth from the 
point A on the days when the different observations were made. 
Let us next take the lines SB, SC, &c, of such lengths that 
each line may be to SA in the inverse ratio of the corresponding 
semi-diameters. If, then, we draw a line through the points 
A, B, C, &c, we shall have a representation of the orbit of the 
earth about the sun. The curve is found to be an ellipse, the 
sun being at one of the foci. The point A, where the earth is 
nearest to the sun, is called the perihelion, the point H, the aphe- 
lion; and the angular distance of the earth from its perihelion 
is called its anomaly. 

The eccentricity of the ellipse, or if is the centre of the 
ellipse, the ratio of OS to OA, is evidently equal to about f^W 
or -g-'oth. A more accurate value of it is .0167917. This eccen- 
tricity is at present subject to a diminution of .000041 a 
century ; but Le Verrier, a French astronomer, has proved that 
after- the eccentricity has diminished to a certain point it will 
begin to increase again. 

THE SEASONS. 

99. The change of seasons on the earth is caused by the 



SEASONS. 



91 



inequality of the days and nights, and this inequality is a result 

of the inclination of the plane of the equinoctial to that of the 

ecliptic. The relative positions of the sun and the earth at dif- 

t ferent parts of the year are represented in Fig. 40. S represents 




Fig. 40. 

the sun, and ABCD the orbit of the earth. Pp is the axis of 
rotation of the earth, and EQ the equator. The plane of this 
equator is supposed to intersect the plane of the ecliptic in the 
line of equinoxes A C. Since, as we have already seen, the sun 
appears to be on this line on the 21st of March and the 21st of 
September, the earth itself must also be on this line at the same 
time. Suppose, then, the earth to be at A on the 21st of March. 
The sun will evidently lie in the direction AS, and will be pro- 
jected on the celestial sphere at the vernal equinox. Now, since 
a line which is perpendicular to a plane is perpendicular to every 
line in that plane which is drawn to meet it, the axis Pp at A, 
being perpendicular to the equator, is also perpendicular to the 
line AS, which is common to both the plane of the equator and 



92 SEASONS. 

the plane of the ecliptic. Half of each parallel of latitude on 
the earth will therefore lie in light and half in darkness ; and 
hence, as the earth rotates on the axis Pp, every point on its 
surface will describe half of its diurnal course in light and 
half in darkness : in other words, day and night will be equal 
over the whole earth. Since the direction of the axis of rota- 
tion remains unchanged, the same condition of things will occur 
when the earth is at C, on the 21st of September. Let the 
earth be at B on the 21st of June. Here we see that, as the 
earth rotates on its axis Pp, every point on its surface within 
the circle ab will lie continually in the light, and will hence 
have continual day, while within the corresponding circle a'U 
the night will be continual. We see also that at the equator 
the days and nights will be equal, and that every point between 
the equator and the circle ab will describe more of its diurnal 
c ourse in light than in darkness, and will thus have its days 
longer than its nights ; while between the equator and the circle 
(ib' the nights will be longer than the days. Similar phenomena 
will occur when the earth is at D, on the 21st of December, 
except only that it will then be the southern hemisphere in 
w r hich the days are longer than the nights, and the southern 
pole at which the sun is continually visible. 

Such, then, is the inequality of the days and nights caused 
by the inclination of the plane of the equinoctial to that of the 
equator. As the sun apparently moves from either equinox, 
the inequality of day and night continually increases, reaches 
its maximum when the sun arrives at either solstice, and then 
continually decreases as the sun moves on to the equinox : the 
day being longer than the night in that hemisphere which is 
on the same side of the equator with the sun. Now, any point 
on the earth's surface receives heat during the day and radiates 
it during the night : and hence, w 7 hen the days are longer than 
the nights, the amount of heat received is greater than the 
amount radiated, and the temperature increases ; while, on the 
contrary, when the days are shorter than the nights, the tempe- 
rature decreases : and thus is brought about the change of rea- 
sons on the earth. 

Another fact, depending on the same cause, and tending to 



SEASONS. 93 

the same result, must also be taken into consideration; and 
that is that the temperature at any place depends on the ob- 
liquity of the sun's rays: on the altitude, in other words, which 
the sun attains at noon. Now we have, from Art. 76, 

z = L — d: 
from which we see that, the latitude remaining constant, the sun 
attains the greater altitude, the greater its declination when it 
has the same name as the latitude, and the less its declination 
when it has the opposite name: so that the nearest approach to 
verticality in the sun's rays will occur at the same time that 
the day is the longest. An exception, however, must be noticed 
to this general rule, in the case of places within the tropics : 
since at these places, as may be seen from the formula, the sun 
passes through the zenith when its declination is equal to the 
latitude, and has the same name. 

100. Effect of the Ellipticity of the Earth's Orbit on the Change 
of Seasons. — The elliptic form of the earth's orbit has very 
little to do with the change of seasons. For although the earth 
is nearer to the sun on the 1st of January than on the 1st of 
July, yet its angular velocity at that time is found by observa- 
tion to be greater, and to vary throughout the whole orbit in- 
versely as the square of the distance. Now it may readily be 
shown that the amount of heat received by the earth at different 
parts of its orbit also varies, other things being equal, inversely 
as the square of the distance : so that equal amounts of heat 
are received by the earth in passing through equal angles of 
its orbit, in whatever part of its orbit those angles may be 
situated. Still, although the change in distance does not mate- 
rially affect the annual change of seasons, it does affect the 
relative intensities of the northern and the southern summer. 
The southern summer takes place when the earth's distance is 
only about ||ths of what it is at the time of the northern 
summer: hence, the intensity at the former period will be to 
that at the latter in the ratio of about (|§) 2 to 1, or about }f to 
1 : in other words, the intensity of the heat of the southern sum- 
mer will be j^th greater than that of the heat of the northern 
summer. 



94 TWILIGHT. 



TWILIGHT. 




101. If the earth's atmosphere did not contain particles of 
dust and vapor, which serve to reflect the rays of light, the 
transition from day to night would be instantaneous, and the 
intermediate phenomenon of twilight would have no existence. 

This phenomenon 
^ is explained in 

Fig. 41, in which 
AB C represents 
a portion of the 
earth's surface, 
Fig. 4i. G and ED F a por- 

tion of the atmosphere. Let the sun be supposed to lie in the 
direction AS, and to be in the horizon of the place A. All of the 
atmosphere which lies above the horizontal plane SD will then 
receive the direct rays of the sun, and A will receive twilight 
from the whole sky. The point B will, on the contrary, be illu- 
minated only by the smaller portion of the atmosphere included 
within the planes EB and AD and the curved surface ED; and 
at the point C the twilight will have wholly ceased. Strictly 
speaking, the lines AS, BE, &c, should be slightly curved, owing 
to the effects of refraction, but the omission involves no change 
in the explanation. 

It is computed that twilight ceases when the sun is about 18° 
below the horizon, measured on a vertical circle. The more 
nearly perpendicular to the horizon is the diurnal circle in 
which the sun appears to move, the more rapid will be the sun's 
descent below the horizon ; hence, the length of twilight dimin- 
ishes as we approach the equator and increases as we recede 
from it. Furthermore, w T e see in Fig. 2 that the greater the 
declination of the sun, the smaller is the apparent diurnal circle 
in which it moves, and the greater will be the length of time 
required for the sun to reach the depression of 18° below the 
horizon. The shortest twilight, therefore, occurs at places on the 
equator, when the sun is on the equinoctial, and its length is 
then lh. 12m. Near the poles the length of twilight is at times 
very great. Dr. Hayes, in his last expedition towards the North 



APPEARANCE OF THE SUN. 95 

Pole, wintered at latitude 78° 18' K, so far above the circle ab, 
Fig. 40, that the sun was continually below the horizon from the 
middle of October to the middle of February ; but at the begin- 
ning and the end of this interval twilight lasted for about nine 
hours. At the poles twilight lasts nearly a month and a half. 

GENERAL DESCRIPTION OF THE SUN. 

102. When the sun is observed with a telescope, spots are 
noticed upon its surface. These spots appear to cross the sun's 
disc from east to west, and with different rates, the rate of 
motion of spots at the sun's equator being the greatest. We 
therefore conclude that these spots have not only an apparent 
motion, caused by the sun's rotation, but also a proper motion 
of their own. By appropriate investigation of these motions, 
it is found that the sun rotates from west to east upon a fixed 
axis, in a plane inclined at an angle of about 7° to the plane of 
the ecliptic. The period of this rotation is about 25 days. 

Much uncertainty exists as to the nature of these spots. The 
generally received opinion is that the sun is surrounded by two 
atmospheres, of which only the outer one (called the photo- 
sphere) is luminous, and that the spots are rents in these 
atmospheres through which the solid body of the sun is seen. 
These spots are for the most part confined to a zone, extending 
about 35° on each side of the sun's equator. They differ widely 
in duration, sometimes lasting for several months, and some- 
times disappearing in the course of a few hours. They are 
sometimes of an immense size. One was seen in 1843, with 
a diameter of nearly 75,000 miles : it remained in sight for 
a week, and was visible to the naked eye. In 1858, a much 
larger one was seen, its diameter being over 140,000 miles. As 
a general thing, each dark spot, or umbra, as it is called, has 
within it a still darker point, called the nucleus, and is sur- 
rounded by a fringe of a lighter shade, called the penumbra. 
Sometimes several spots are inclosed by the same penumbra; 
and occasionally spots are seen without any penumbra at all. 

On the theory of two atmospheres, the existence of the 
penumbra is explained by supposing the aperture in the outer 
and luminous stratum to be wider than that in the inner one, 



96 APPEARANCE OF THE SUN. 

find that portions of the inner stratum, being subjected to a 
strong light from above, are rendered visible : the umbra itself 
heing, as already remarked, the solid body of the sun seen through 
both strata. According to Mr. J. N. Lockyer, "sun-spots are 
cavities or hollows eaten into the photosphere, and these different 
shades [the penumbra, umbra, and nucleus] represent different 
depths." 

One very curious and interesting discovery in relation to these 
spots is that of a periodicity in their number. This discovery 
was made by Schwabe, of Dessau, whose researches and obser- 
vations on this subject covered a period of more than twenty-five 
years. The number of groups of spots which he observed in a 
year varied from 33 to 333, the average being not far from 150.* 
He found the period from one maximum to another to be about 
ten years. Professor Wolf, of Zurich, after tabulating all the ob- 
servations of spots since 1611, decided that the period varied from 
eight to sixteen years, its mean value being about eleven years. 
Recent investigations show that this periodicity is in some way 
connected with the action of the planets, of Jupiter and Venus 
particularly, upon the sun's photosphere. It is a curious fact that 
magnetic storms and the phenomenon called Aurora or Northern 
Lights have a similar period, and are most frequent and most 
striking when the number of the solar spots is the greatest. 

Still other phenomena which are seen upon the sun's disc are 
the facnlce, which are streaks of light seen for the most part in 
the region of the spots, and which are undoubtedly elevations or 
ridges in the photosphere: and the luculi, which are specks of 
light scattered over the sun's disc, giving it an appearance not 
unlike that of the skin of an orange, though relatively much less 
rough. The cause of these luculi is unknown. 

At the time of a total eclipse of the sun by the moon, the disc 
of the sun is observed to be surrounded by a ring or halo of light, 
which is called the corona. The breadth of this corona is more 
than equal to the diameter of the sun. Many theories have been 
advanced to explain this phenomenon, one of which is that it is 
due to the existence of still another atmosphere, exterior to the 

* A table of Schwabe's observations is given in the Appendix. 



CONSTITUTION OF THE SUN. 97 

photosphere. Another theory is that this corona consists of 
streams of luminous matter, radiating in all directions from the 
sun. Rose-colored protuberances, sometimes called red flames, 
are also seen, which are usually of a conical shape, and are 
sometimes of great height. In the total eclipse of August 17th, 
1868, one was observed with an apparent altitude of 3', corre- 
sponding to a height of about 80,000 miles. These protuberances 
were formerly supposed to be similar in character to our terres- 
trial clouds; but Dr. Jannsen, the chief of the French expedi- 
tion sent out to the East to observe the total eclipse of August, 
1868, examined their light with the spectroscope, and found them 
to be masses of incandescent gas, of which the greater part was 
hydrogen. Dr. Jannsen also made the interesting discovery that 
these protuberances can be examined at any time, without wait- 
ing for the rare opportunity afforded by a total eclipse. He 
observed them for several successive days, and found that great 
changes took place in their form and size. Mr. Lockyer, of 
England, who has since examined Jhem, pronounces them to be 
merely local accumulations of a gaseous envelope completely 
surrounding the sun : the spectrum peculiar to these protuber- 
ances appearing at all parts of the disc. 

It has already been stated (Art. 47) that the spectroscope en- 
ables us to establish the existence of certain chemical substances 
in the sun, by a comparison of the spectra of these substances 
with that of the sun ; or, more precisely, by a comparison of the 
lines, bright or dark, by which these different spectra are dis- 
tinguished. The number of the parallel dark lines in the solar 
spectrum which have been detected and mapped exceeds 3000; 
and careful examination also shows that some of these are double. 
Some of the more prominent of these lines have received the 
names of the first letters of the alphabet; D, for example, is a 
very noticeable double line in the orange of the spectrum. When 
certain chemical substances are evaporated, either in a flame or 
by the electric current, the spectra which they form are also 
characterized by lines, which, however, are not dark, but bright. 
If, for instance, sodium is introduced into a flame, its incan- 
descent vapor produces a spectrum which is characterized by a 
brilliant double band of yellow; and it is especially noticeable 

7. 



08 CONSTITUTION OF THE SUN. 

that this yellow band coincides exactly in position with the 
dark line D of the solar spectrum. In the same way the 
spectrum of zinc is found to contain bands of red and blue- 
that of copper contains bands of green: and, in general, the 
spectrum of each metal contains certain bright bands or lines, 
peculiar to itself, and readily recognized. We may therefore 
conclude that an incandescent gas or vapor emits rays of a certain 
refrangibility and color, and those rays only. 

Again, it is proved by experiment that if a ray of white light 
be allowed to pass through an incandescent vapor, the vapor 
will absorb precisely those rays which it can itself emit If, for 
instance, a continuous spectrum be formed by a ray of intense 
white light from any source, and if the vapor of sodium be intro- 
duced in the path of this ray, between the prism and the source 
of light, a dark band will appear in the spectrum, identical in 
position with the bright yellow band which we have already 
noticed in the spectrum of sodium, and which we found to be 
identical in position with the dark line D of the solar spectrum. 

We are now ready to apply the principles established by 
these experiments to the case of the sun. The sun is, as we 
saw above, a sphere surrounded by a vaporous envelope. This 
sphere would of itself emit all kinds of rays, and therefore 
give a continuous spectrum ; but the photosphere which sur- 
rounds it absorbs those of the sun's rays which it can itself 
emit. The dark line- D of the solar spectrum shows, as in the 
experiment above described, that sodium has been introduced 
in the path of the sun's rays : in other words, that sodium is in 
the sun's photosphere. In the same way, Professor Kirchhoff, 
to whom we owe this remarkable discovery, has established the 
existence in the photosphere of iron, calcium, magnesium, chro- 
mium, and other metals. In the case of iron, more than sixty 
bright lines have been detected in its spectrum : and for every 
one of these lines there is a corresponding dark line in the solar 
spectrum. 

We also see, from the preceding experiments, how the presence 
of bright lines in the spectrum of the rose-colored protuberances 
could prove to Dr. Jannsen that these protuberances were not 
masses of clouds, reflecting the light of the sun, but masses of 



ZODIACAL LIGHT. &9 

incandescent vapor. We shall see another instance of the saruc 
description when we come to examine some of the nebulse. 

THE ZODIACAL LIGHT. 

103. At certain seasons of the year a faint nebulous light. 
not unlike the tail of a comet, is seen in the west after twilight 
has ended, or in the east before it has begun. This is called 
the Zodiacal Light. Its general shape is nearly that of a cone, 
the base of which is turned towards the sun. The breadth of 
the base varies from 8° to 30° of angular magnitude. The apex 
of the cone is sometimes more than 90° to the rear or in advance 
of the sun. According to Humboldt, it is almost always visible, 
at the times above stated, within the tropics : in our latitudes it 
is seen to the best advantage in the evening near the first of 
March, and in the morning near the middle of October. 

Of the many theories proposed to account for the zodiacal 
light, the one which seems to be most widely accepted is that it 
consists of a ring or zone of rare nebulous matter encircling 
the sun, which reaches as far as the earth, and perhaps extends 
beyond it. According to another theory, it is a belt of meteonV 
bodies surrounding the sun. A very interesting and valuable 
series of observations upon the Zodiacal Light was made by 
Chaplain Jones, United States Navy, in the years 1853-5, in 
latitudes ranging from 41° N. to 53° S. The conclusion which 
he drew from his observations was that the light was a nebulous 
ring encircling the earth, and lying within the orbit of the moon. 



100 SIDEREAL AND SOLAR TIMES. 



CHAPTER VII. 

SIDEREAL AND SOLAR TIME. THE EQUATION OF TIME. THE 

CALENDAR. 

104. Sidereal and Solar Days. — It is important to distin- 
guish between the apparent annual motion of the sun in the 
ecliptic, from west to east, and the apparent diurnal motion 
towards the west, which the rotation of the earth gives to all 
celestial bodies and points. A sidereal day is the interval of 
time between two successive transits of the vernal equinox over 
the same branch of the meridian. A solar day is the interval 
between two similar transits of the sun. But the continuous 
motion of the sun towards the east causes it to appear to move 
more slowly towards the west than the vernal equinox moves. 
The solar day is therefore longer than the sidereal day, the 
average amount of the difference being 3m. 55.5s. And fur- 
thermore, in the interval of time in which the sun makes one 
complete revolution in the ecliptic, the number of daily revo- 
lutions which it appears to make about the earth will be less 
by one than the number of daily revolutions made by the 
equinox. The sidereal year, then (Art. 89), which contains 
365d. 6h. 9m. 9.6s. of solar time, contains 366d. 6h. 9m. 9.6s. 
of sidereal time. 

105. Relation of Sidereal and Solar Times. — Since the side- 
real day is shorter than the solar day (and, consequently, the 
sidereal hour, minute, &c, than the solar hour, minute, &c), it 
is evident that any given interval of time will contain more 
units of sidereal than of solar time. The relative values of the 
sidereal and the solar days, hours, &c, are obtained as follows: — 
We have from the preceding article, 

366.25636 sidereal days = 365.25636 solar days : 
one sidereal day = 0.99727 solar day, 
one sidereal hour ==« 0.99727 solar hour, &c. 



EQUATION OF TIME. 10! 

Having, therefore, an interval of time expressed in either solat* 
or sidereal units, we may easily express the same interval hi 
units of the other denomination. This is called the conversion 
of a solar into a sidereal interval, and the reverse : and tables 
for facilitating this conversion are given in the Nautical Al- 
manac. 

Again, knowing the sidereal time at any instant, the hour- 
angle, that is to say, of the vernal equinox, the corresponding 
solar time, or the hour-angle of the sun, is readily obtained by 
subtracting from the sidereal time the sun's right ascension. 
This is indeed a corollary of the theorem proved in Art. 9, 
from which we see that the sum of the sun's right ascension 
(which can always be found in the Nautical Almanac), ami 
its hour-angle, is the sidereal time. Either of these times, then, 
may be converted into the other. 

THE EQUATION OF TIME. 

106. Inequality of Solar Days. — Observation shows that the 
length of the solar day is not a constant quantity, but varies at 
different seasons of the year, and, indeed, from day to day. A 
distinction must therefore be made between the apparent or 
actual solar day, and the mean solar day, which is the mean 
of all the apparent solar days of the year. A uniform measure 
of time may be obtained from the apparent diurnal motion 
with reference to our meridian of a fixed celestial body or 
point. It may also be obtained from the apparent diurnal 
motion of a celestial body which changes its position in the 
heavens, provided that two conditions are satisfied; first, the 
plane in which the body moves must be perpendicular to the 
plane of the meridian: and second, its motion in that plane 
must be uniform. Both these conditions are so very nearly 
satisfied by the motion of the vernal equinox, that any two 
sidareal days may be considered to be sensibly equal to each 
other; but neither condition is satisfied by the motion of the 
sun. It moves in the ecliptic, the plane of which is not, in 
general, perpendicular to the plane of the meridian : and its 
motion in this plane is not uniform. We have, therefore, two 




102 EQUATION OF TIME. 

causes of the inequality of the solar days, the effect of each of 
which we will now proceed to examine. 

107. Irregular Advance of the Sun in the Ecliptic. — Observa- 
tion show r s that the sun's motion in longitude is not uniform. 
The mean daily motion is, of course, obtained by dividing 360° 
by the number of days and parts of a day in a year, and is 
59' 8."3. But the daily motion about the first of January is 
61' 10", while about the first of July it is only 
57' 12". In Fig. 42, let the circle A MM" 
represent the apparent orbit of the sun in 
the ecliptic about the earth E, and let the 
sun be supposed to be at the point A w 7 here 
its daily motion is the greatest, on the first 
of January. Let us also suppose a fictitious 
sun (which we w r ill call the first mean sun) 
to move in the ecliptic with the uniform rate of 59' 8". 3 daily, 
and to be at the point A at the same time that the true sun is 
there. On the next day the mean sun will have moved eastward 
to some point M f while the true sun, whose daily motion is at this 
time greater than that of the mean sun, will be found at some 
point T, to the east of M. The true sun will continue to gain on 
the mean sun for about three months, at the end of which time 
the mean sun will begin to gain on the true sun, and will finally 
overtake it at the point B, on the first of July. During the 
second half of the year the mean sun will be to the east of the 
true sun, and at the end of the year the two suns will again be 
together at A. 

The angular distance between the tw*o suns, represented in 
the figure by the angles TEM, T'EM\ &c, is called the Equa- 
tion of the Centre. It is evidently additive to the mean longi- 
tude of the sun while it is moving from A to JS, and subtractive 
from it while it is moving from B to A. Its greatest value is 
about 8 minutes of time. 

Since the rotation of the earth gives to both these bodies a 
common daily motion to the west, it is plain that from January 
to July the mean sun will cross the meridian before the true 
sun, and that from July to January the true sun will cross the 
meridian before the mean sun. 



/ ctM- 


E 


\f 


\ 






.V? \ 


\ 3-rl I 








\S[ m 1 1 








\ b c d 


e 


f 


9 J 



EQUATION OF TIME. 103 

108. Obliquity of the Ecliptic to the Meridian. — Even if the 
sun's motion in the ecliptic were uniform, equal advances of the 
sun in longitude would not be accompanied by equal advances 
in right ascension, in consequence of the obliquity of the 
ecliptic to the meridian. The truth of this may be seen in 
Fig. 43. Let this figure represent P 

the projection of the celestial sphere 
on the plane of the equinoctial co- 
lure PApH. A and H are the 
equinoxes, P and p the celestial 
poles, AeH the equinoctial, and 
A EH the ecliptic. Let the ecliptic 
be divided into equal arcs, AB, EC, 
&c, and through the points of divi- 
sion, B, C, &c, let hour-circles be p 

. . . . Fig. 43. 

drawn, meeting the equinoctial in 

the points b, c, &c. Now, since all great circles bisect each 
other, AEH is equal to AeH, and if Pep is the projection of an 
hour-circle perpendicular to the circle PApH, AE and Ae are 
quadrants, and equal. The angle PBC is evidently greater 
than PAB, PCD is greater than PBC, &c. : in other w T ords, the 
equal arcs AB, BC, &c, are differently inclined to the equi- 
noctial. The effect of this is that the equinoctial is divided 
into unequal parts by the hour-circles Pb, Pc, &c, be being 
greater than Ab, cd than be, &c. It is to be noticed, further, 
that the points B and b, being on the same hour-circle, will be 
on the meridian at the same instant of time: and the same is 
true of C and c, D and d, &c. 

Now, if A is the vernal equinox, the first mean sun, moving 
in the ecliptic with the constant daily rate of 59' 8".3, will 
pass through that point on the 21st of March. Let another 
fictitious sun (called the second mean sun) leave the point A at 
the same time, and move in the equinoctial with the same uni- 
form daily rate. Since BAb is a right-angled triangle, Ab is 
less than AB. Hence, when the first mean sun reaches B, the 
second mean sun will be at some point m, to the east of b: 
when the first mean sun is at C, the second mean sun will be to 
the east of c, &c. : and the second mean sun will continue to 



104 EQUATION OF TIME. 

lie to the east of the first mean sun until the 21st of June (the 
summer solstice), when both suns will be at the points E and e 
at the same instant of time, and will therefore come to the 
meridian together. In the second quadrant, the second mean 
sun will lie to the west of the first mean sun, and both suns 
will reach jET, the autumnal equinox, on the 21st of September. 
The relative positions in the third and the fourth quadrant will 
be identical with those in the first and the second. 

From the 21st of March, then, to the 21st of June, the second 
mean sun, being to the east of the first mean sun, will come 
later to the meridian ; and the same will also be true from the 
21st of September to the 21st of December. In the two other 
similar periods the case will be reversed, and the second mean 
sun will come earlier to the meridian than the first mean sun. 
The greatest difference of the hour-angles of these two mean suns 
is about 10 minutes of time. 

109. Equation of Time. — It is by means of these two fictitious 
suns that we are able to obtain a uniform measure of time from 
the irregular advance of the sun in the ecliptic. The second 
mean sun satisfies the two conditions stated in Art. 10G, and 
therefore its hour-angle is perfectly uniform in its increase. 
This hour-angle is the mean solar time of our ordinary watches 
and clocks. The hour-angle of the true sun is called the ap- 
parent solar time: and the difference at any instant between the 
apparent and the mean solar time is called the equation of time. 

Let Fig. 44 be a projection of the 
celestial sphere on the plane of the 
horizon. Z is the zenith, P the 
pole, EVQ the equinoctial, CL the 
ecliptic, and Fthe vernal equinox. 
Let T be the position of the true 
sun in the ecliptic, and M that of 
second mean sun in the equinoctial. 
The angle TPM is evidently the 
equation of time. This angle is mea- 
sured by the arc AM, or VM — VA : 
the difference, that is, of the right ascensions of the true and 
the second mean sun. But since the angular advance of the 




CALENDAR. 105 

second mean sun in the equinoctial is, by hypothesis, as shown 
in the previous article, equal to the angular advance of the first 
mean sun in the ecliptic, it follows that the right ascension of 
the second mean sun is always equal to the longitude of the 
first mean sun, or, as it is usually called, the true sun's mean 
longitude. The equation of time, then, is the difference of the 
suns trite right ascension and mean longitude; and thus com- 
puted is given in the Nautical Almanac for each day in the 
year. It reduces to zero four times in the year, and passes 
through four maxima, ranging in value from 4 minutes to 16 
minutes. 

110. Astronomical and Civil Time. — The mean solar day is 
considered by astronomers to begin at mean noon, when the 
second mean sun (usually called simply the mean sun) is at its 
upper culmination. The hours are reckoned from Oh. to 24h. 
The mean solar day, so considered, is called the astronomical day. 

The civil day begins at midnight, twelve hours before the 
astronomical day, and is divided into two parts of twelve hours 
each, called a.m. and p.m. 

We must, therefore, carefully distinguish between any given 
civil time and the corresponding astronomical time. For in- 
stance, January 3d, 8 a.m., in civil time, is the same as Janu- 
ary 2d, 20h., in astronomical time. 

THE CALENDAR. 

111. Owing to causes which will be explained further on, the 
position of the vernal equinox is not absolutely stationary, but 
moves westivard along the ecliptic, with an annual rate of about 
50". 2. The sun, then, moving eastward from the equinox, will 
reach it again before it has made one complete sidereal revo- 
lution about the earth. This interval of time in which the sun 
moves from and returns to the equinox is called a tropical year, 
and consists of 365d. oh. 48m. 47.8s. The Julian Calendar 
was established by Julius Gesar, 44 B.C., and by it one day was 
inserted in every fourth year. This was the same thing as as- 
suming that the length of the tropical year was 365d. 6h., 
instead of the value given above, thus introducing an accumu- 
lative error of 11m. 12s. every vear. This calendar was 



100 CALENDAR. 

adopted by the Church in 325 a.d., at the Council of Nice, and 
the vernal equinox then fell on the 21st of March. In lf)82, 
the annual error of 11m. 12s. caused the vernal equinox to fall 
on the 11th of March, instead of the 21st. Pope Gregory XIII. 
therefore ordered that ten days should be omitted from the year 
1582, and thus brought the vernal equinox back again to the 
21st of March. Furthermore, since the error of 11m. 12s. a year 
amounted to very nearly three days in 400 years, it was decided 
to leave out three of the inserted days (called intercalary days) 
every 400 years, and to make this omission in those years which 
were not exactly divisible by 400. Thus of the years 1700, 
1800, 1900, 2000, all of which are leap years according to the 
Julian calendar, only the last is a leap year according to the 
reformed or Gregorian calendar. By this calendar the annual 
error is only 24 seconds, and will not amount to a day in much 
less than 4000 years. 

This reformed calendar was not adopted by England until 
1752, when eleven days were omitted from the calendar. The 
two calendars are now often called the old style and the new style. 
For instance, April 26th, O.S., is the same as May 8th, N.S. In 
Russia the old style is still retained, though it is customary to 
give both dates ; as 1868, ^~. All other Christian countries 
have adopted the new style. 



UNIVERSAL GRAVITATION. 107 



CHAPTER VIII. 

LAW OF UNIVERSAL GRAVITATION. PERTURBATIONS IN THE 
EARTH'S ORBIT. ABERRATION. 

112. The Lata of Universal Gravitation. — The earth, as we 
have seen in Chapter VI., revolves about the sun in an elliptical 
orbit, with a linear velocity of eighteen miles a second. At every 
point of its orbit the centrifugal force induced by this revolution 
must create in the earth a tendency to leave its orbit, and to go 
off in the direction of a tangent to the orbit at that point. To 
counteract this centrifugal force, there must constantly exist a 
centripetal force, by which the earth is at every instant deflected 
from this rectilinear path which it tends to follow, and is drawn 
towards the sun; and in order that the orbit of the earth may 
remain unchanged in form, — as observation shows that it does 
remain,— these two forces must be in constant equilibrium. Ad- 
mitting, then, the existence of such a centripetal force, it remains 
to determine the nature of the force, and the laws under which 
it acts. 

The force is believed to be identical in nature with that force 
which causes all bodies, free to move, to tend towards the earth's 
centre, and which we call the force of gravity. At whatever 
height above the surface of the earth the experiment may be 
made, this attractive force of the earth is found to exist ; and 
there is no good reasdn for assuming any finite limit beyond 
which this force, however much its effects may be lessened by 
other and opposing forces, does not have at least a theoretic 
existence. And, furthermore, as the sun and the other heavenly 
bodies are all masses of matter like the earth, there is every 
reason for concluding that they too, as well as the earth, possess 
this power of attracting other bodies towards their centres. Nor 
is this attractive power a characteristic of large bodies alone: for 



108 UNIVERSAL GRAVITATION. 

we have already seen in the experiment with the torsion balance, 
described in Art. 66, that small globes of lead exert a sensible 
attraction upon still smaller globes. We may therefore assume 
that what is true of each of these masses, large and small, as a 
whole, is no less true of the separate particles of which it is com- 
posed, and that every particle of matter in the universe has an 
attractive power upon every other particle. 

In order to determine the laws under which this attractive 
power is exerted, we have only to assume that the laws which 
are shown by experiment to obtain at the earth's surface hold 
equally good throughout the universe; so that whatever the 
masses of bodies may be, or whatever the distances by which 
they are separated from each other, the forces with which any 
two bodies attract a third will be directly proportional to the 
masses of the two attracting bodies, and inversely proportional 
to the squares of their distances from the third body. 

This, then, is Newton's Law of Universal Gravitation. Every 
particle of matter in the universe attracts every other particle, 
with a force directly proportional to the mass of the attracting 
particle, and inversely proportional to the square of the distance 
between the particles. In applying this general law to the par- 
ticles which compose the masses of the heavenly bodies, Newton 
has demonstrated that the attraction exerted by a sphere is pre- 
cisely what it w T ould be if all the particles in the sphere were 
collected at its centre, and constituted one particle, with an at- 
tractive power equal to the sum of the powers of these different 
particles. 

113. Verification of the Law in the Case of the Moon. — The 

moon is shown by observation to revolve about the earth in a 

period of 27.32 days, at a mean distance from the earth of 238,800 

miles. If we take the formula for centrifugal force given in 

Art. 69, 
l 4- 2 r 

and substitute for r the moon's distance in feet, and for t its 
period of revolution in seconds, we shall find for the centrifugal 
force, 

/ = 0.0089 feet: 



MASS OF THE SUN. 109 

that is to say, in one second the earth tends to give the moon a 

velocity towards itself of 0.0089 feet. Now the force of gravity 

on the eartns surface at the equator is 32.09 feet; and if the law 

of gravitation is assumed to be true, the force of gravity at the 

32 09 
distance of the moon will be — feet, since the distance of 

(60.267) 2 

the moon from the earth is equal to 60.267 of the earth's radii. 
The value of this expression is found to be 0.0088 feet. The two 
results vary by only yooo^h °f a foot: and it is therefore fair to 
conclude that the centrifugal force of the moon in its orbit is 
really counteracted by the earth's attraction. 

In whatever way the law of gravitation is tested in connection 
with the observed motions of the heavenly bodies, the facts which 
come by observation are alw T ays found to be in close agreement 
with the results which the law demands ; and it is safe to say 
that the truth of this law is as satisfactorily demonstrated as is 
that of the laws of refraction, of the laws of sound, or of the 
many other natural laws which depend upon observation and 
experiment for their ultimate proof. 

114. The Mass of the Sun. — Let A denote the attraction ex- 
erted by the sun on the earth, and a that exerted by the earth 
on a body at its surface. Let M denote the mass of the sun, m 
that of the earth, r the radius of the earth, and R the radius 
of the earth's orbit. We have, then, by the law of gravitation, 
A M r 2 
~a~~ ~^T X W 
But A must equal the earth's centrifugal force in its orbit, or 

4- 2 i? 

— -w- > in which t is 365.256 days, reduced to seconds, and R is 

expressed in feet. We have also a equal to 32.09 feet. Substi- 
tuting these values, we have, 

_M_ 4- 2 jR 3 
m ~ 32.092V 2 ' 
Substituting the known values of the different quantities, we 
shall have 

— = 327,000: 
m 

or the mass of the sun is equal to that of 327,000 earths. 



110 MOTION OF THE EARTH. 

The density of the sun, compared with that of the ea./C*, Kehip 

327/ahi 
equal to the mass divided by the volume, is T~oop :,«/ The 

density is therefore equal to about Hh of that of the earth, and 
to about |ths of that of water. 

115. Weight of Bodies at the Surface of the Sun.— The weight 
of the same body at the surface of the earth and ai that of the 
sun will be directly as the masses of the two spheres a*.d inversely 
as the squares of their radii. We shall find thav the weight 
of a body at the sun is about 27.7 times its weight h,t the earth ; 
so that a body which exerts a pressure of 10 pounds at the earth 
w r ould exert a pressure at the sun equal to that ot 277 of the 
same pounds; and a man whose weight is 150 pounds would, 
if transported to the sun, be obliged to support in his own body 
a weight equivalent to about two of our tons. 

116. The Earth's Motion at Perihelion and Aphelion. — We 
have already seen that the angular velocity of the earth in its 
orbit is the greatest at perihelion, when the earth is the nearest to 
the sun, and is the least at aphelion, when the earth is the farthest 
from the sun. This irregularity of motion is a consequence of 
the attraction exerted by the sun on the earth, as may be seen in 

~ D Fig. 45. In this figure, let S be the sun, 
P the perihelion of the earth's orbit, and 
A the aphelion. Let the earth move 
from A to P, and suppose it to be at the 
point E. The attraction of the sun on 
the earth, along the line ES, may be re- 
solved into two forces, one of w T hich, 
acting in the direction of the tangent EB, 
will evidently tend to increase the velo- 
Fi g . 45. city of the earth in its orbit. At the 

point E", on the contrary, where the earth is moving toward A, 
the effect of the sun's attraction is to diminish the earth's velocity. 
In general, then, the earth's velocity will increase as it moves 
from aphelion to perihelion, and decrease as it moves from peri- 
helion to aphelion. 




kepler's laws. Ill 



KEPLER S LAWS. 



117. In the early part of the seventeenth century, more than 
fifty years before the announcement by Xewton of the law of 
universal gravitation, the astronomer Kepler, by an examination 
of the observations which had been made upon the motions of the 
planets, and which had shown that the planets revolved about 
the sun, discovered the following laws: — 

(1.) The orbit of every planet is an ellipse, having the sun at 
one of its foci. 

(2.) If a line, called a radius vector, is supposed to be drawn 
from the sun to any planet, the areas described by this line, as 
the planet revolves in its orbit, are proportional to the times. 

(3.) The squares of the times of revolution of any two planets 
are proportional to the cubes of their mean distances from the sun. 

These laws were verified by Kewton in his Prlncipia, in a 
course of mathematical reasoning, the foundation of which was 
his theory of universal gravitation. With regard to Kepler's 
first law, he showed that any two spherical bodies, mutually at- 
tracted, describe orbits about their common centre of gravity, 
and that these orbits are limited in form to one or another of the 
four conic sections, — the circle, the ellipse, the parabola, and the 
hyperbola. For instance, in the case of the earth and the sun, 
the earth does not describe an ellipse about the sun at rest, but 
both earth and sun revolve about their common centre of gravity. 
It is shown in Mechanics that the common centre of gravity of 
any two globes is at a point on the straight line joining their in- 
dependent centres of gravity, so situated that its distances from 
the centres of the two globes are inversely as the masses of the 
globes. Hence the distance of the common centre of gravity 

**i .u a.x, e \x. ru. • 92,400,000 
ot the earth and the sun from the centre ot the sun is — ' 

327,000 
miles, or only about 280 miles; so that the sun may practically 
be considered to be at rest. 

With regard to Kepler's second law, Newton further showed 
that the angular velocity with which the radius vector moves 
must be inversely proportional to the square of its length. 



112 PRECESSION. 

Finally, be proved also the truth of the third law, provided 
only that a slight correction is introduced when the mass of the 
planet is not so small as to be inappreciable in comparison with 
that of the sun. If t and t' denote the times of revolution of 
any two planets, m and m! their masses (the mass of the sun 
being unity), and d and df their mean distances from the sun, 
Newton showed that we shall always have the following pro- 
portion : 

d? d' 3 

P:tf* = 



1 -j- m ' 1 -\-m! 

If m and m' are so small that they may be omitted without 
sensible error, this proportion becomes identical with Kepler's 
third law. 

PERTURBATIONS IN THE EARTIl's ORBIT. 

118. Precession. — Although the absolute positions of the 
planes of the ecliptic and the equinoctial in space, and their re- 
lative positions to each other, remain very nearly the same from 
year to year, there are nevertheless certain small perturbations 
in these positions which are made evident to us by refined and 
extended observations. The principal of these perturbations is 
called precession. 

The latitudes of all the fixed stars remain very nearly the 
same from year to year, and even from century to century: and 
we therefore conclude that the position of the ecliptic with 
reference to the celestial sphere remains very nearly unchanged. 
But the longitudes of the stars are all found to increase by an 
annual amount of 50".2: and hence the line of the equinoxes 
must have an annual westivard motion of the same amount. 
This westward motion is called the precession of the equinoxes. 
Since the ecliptic remains stationary in the heavens (or at least 
so nearly stationary that the latitudes of the stars only vary 
by half a second of arc in a year), this precession must be con- 
sidered to be a motion of the equinoctial on the ecliptic. 

In Fig. 46, let EQ represent the equinoctial, and LC the eclip- 
tic. BVis the line of the equinoxes, which moves about in the 
plane of the ecliptic, taking in course of time the new position 
.CF'. Perhaps the clearest conception of this motion is ob- 



PRECESSION. 






tained by considering P, the pole of 
the equinoctial, to revolve about A, 
the pole of the ecliptic, in the circle 
PG (the polar radius of which, AP, 
is 23° 27'), moving westward in this 
circle with the annual rate of 50". 2, 
and completing its revolution in 
25,817 years. 

A general explanation of the 
cause of precession may be given by 
means of Fig. 47. The earth may 
be regarded as a sphere surrounded by a spheroidal shell, as 
represented in the 
figure by EPQp, 
and the matter in 
this shell may be 
considered to form 
a ring about the 
earth in the plane 
of the equator, as 
shown in EP'Q'p'. 
It is to the attrac- 
tion of the sun and 
the moon on this 
ring, combined with 
the earth's rotation, 
that the precession 
of the equinoxes is 
due. In the figure 

let S be the sun, the Fi s- 47 - 

circle AB V the ecliptic, and E" ' Q" this equatorial ring of the 
earth. Let ACVbe the plane of the equinoctial, meeting the 
plane of the ecliptic in the line of equinoxes A F. This plane 
is by definition determined at each instant by the position of 
the earth's equator. The attraction exerted by the sun upon 
the different particles of the ring in that half of it which is 
nearer the sun (the particle E'\ for instance), may be resolved 
into two forces, one acting in the plane of the equator, and the 




1 1 1 PRECESSION. 

Dther in a direction perpendicular to that plane, or in the direc- 
tion E"d. The sun's attraction upon the nearer half of the 
ring, then, tends to draw the plane of the ring nearer to the 
plane of the ecliptic. On the other hand, the sun's attraction 
upon the farther half of the ring tends to bring about the 
opposite result ; but since, by the law of attraction, the latter 
effect is less than the former, we may consider the whole result 
of the sun's attraction upon the ring to be a tendency in the 
plane of the ring to come nearer to the plane of the ecliptic. 
Therefore, if the ring did not rotate, the plane of the earth's 
equator would ultimately come into coincidence with the plane 
of the ecliptic. 

But the ring does rotate, about an axis perpendicular to its 
own plane ; and the combined result of this rotation and of the 
rotation about the line of the equinoxes, above described, is 
that the plane of the equinoctial, while it preserves constantly 
its inclination to the plane of the ecliptic, moves about in a 
westerly direction: the line of intersection of the two planes 
also moving about in the same direction, and thus giving rise to 
the precession of the equinoxes. 

Similar results will evidently follow if S represents the moon 
instead of the sun. Owing to the greater proximity of the 
moon to the earth, however, the results of its attraction are 
more than double those of the attraction of the sun. There is 
still another perturbation in the position of the line of the equi- 
noxes which is a result of the mutual attraction between the 
earth and the other planets. This attraction tends to draw the 
earth out of the plane of the ecliptic, without affecting in any 
way the position of the plane of the equinoctial. The result is 
an annual movement of the equinoxes towards the east. This 
perturbation is exceedingly minute, being only about ^th of a 
second of arc in a year. The value 50". 2 is the algebraic sum 
of all these perturbations. 

119. Results of Precession. — One result of precession is to 
make the interval of time between two successive returns of the 
sun to the vernal equinox less than the time of one sidereal 
revolution, by the time required by the sun to pass over 50".2, 
which is 20m. 21.8s. Hence we have the tropical year, to which 



NUTATION. 115 

reference has already been made in Art. 111. Another result 
is that the signs of the Zodiac (Art. 91) no longer coincide with 
the constellations after which they are named, but have retreated 
towards the west by about 28°, or nearly one sign : so that the 
constellation of Aries now lies in the sign of Taurus. Still 
another result is that the same star is not the pole-star in dif- 
ferent ages. Referring to Fig. 46, the pole of the heavens, P, 
will have revolved about A to the position G, in the course of 
about 13,000 years; and a star of the first magnitude, called 
Vega, which is now about 51° from the pole, will at that time be 
less than 5° from the pole, and will be the pole-star. 

120. Nutation. — Since precession is the result of the tendency 
of the sun to change the position of the plane of the equator, it 
is evident that there will be no precession when the sun is itself 
in the plane of the equator, — in other words, at the equinoxes, — 
and that the precession will be at its maximum when the sun is 
the farthest from the plane of the equator: that is to say, is at the 
solstices. The amount of precession due to the influence of the 
moon is subject, to a similar variation, being the greatest when 
the moon's declination is the greatest. The result is that the 
pole of the heavens has a small oscillatory motion about its 
mean place. This motion is called nutation. If the effect of 
nutation could be separated from that of precession, the pole 
would be found to move in a very minute ellipse, having a 
major axis of 18". 5, and a minor axis of 13". 7, the period of 
one revolution in this ellipse being about nineteen years. Since, 
however, these two perturbations co-exist, the result is that the 
pole of the heavens revolves about the pole of the ecliptic, not 
in a circle, but in an undulating curve, as 
represented in Fig. 48 : the amount of the 
deviation being very much exaggerated in 
the figure. 

121. Change in the Obliquity of the Eclip- 
tic. — It was stated in Art. 118 that the lati- 
tudes of the stars were found to vary from IST^T 
year to year by a very minute amount. 
This change in the latitudes is due to a change in the position 
of the plane of the ecliptic, involving a change in the obliquity 




116 ABERRATION. 

of the ecliptic. The obliquity of the ecliptic in 1868 was 
23° 27' 23": and the annual amount of diminution to which it 
is now subject is 0".46. Mathematical investigations show that 
after certain moderate limits have been reached this diminution 
will cease, and the obliquity will begin to increase. The arc 
through which the obliquity oscillates is about 1° 21', and the 
time of one oscillation is about ten thousand years. 

122. Advance of the Line of Apsides. — The line connecting 
the earth's perihelion and aphelion is called the line of apsides. 
This line revolves from west to east, with an annual rate of 11". 8; 
a perturbation due to the attraction exerted on the earth by the 
superior planets. The time in which the earth moves from peri- 
helion to perihelion is called the anomalistic year (from anomaly, 
Art. 98). It is evidently longer than the sidereal year, and is 
found to contain 365d. 6h. 13m. 49.3s. 

ABERRATION. 

123. The apparent direction of a celestial body is determined 
by the direction of the telescope through which it is observed. 
In consequence of the motion of the earth, and the progressive 
motion of light, the telescope is carried to a new position while 
the light is descending through it, and therefore the apparent 
direction of the body will differ from its true direction. 

\? In Fig. 49, let OF be the posi- 

$p \ tion of the axis of a telescope at 

the instant when the rays of light 
from the star 8 reach the object 
glass 0. The rays, after passing 
through the glass, begin to con- 
verge towards a fixed point in 
space, with which, at this instant, 
the intersection of the cross-wires 
Fig. 49. coincides. Let the earth be moving 

in the direction FA. Since the transmission of light is not 
instantaneous, time is required for the light to pass from to 
the fixed point in space, and in that time the earth will carry 
the axis of the telescope to some new T position OF f . The cross- 
wires will then be at F\ while the rays, whose motion in space 




ABEBjBATION. 117 

is en f irely independent of any motion of the telescope, will tend 
to meet at the point jP. In order, then, to have the image of the 
star coincide with the intersection of the wires, the telescope 
must be so moved that its axis will lie in the position OF. The 
star will then appear to lie in the direction FS', while its true di- 
rection is of course FS : and the angle w T hich these two directions 
make with each other, or the angle F f O'F, is called the aberra- 
tion. Representing this angle by A, and the angle OFF' ', the 
angle between the apparent direction of the star and the direc- 
tion in which the earth is moving, by I, we shall have, 

sin .4; sin I=FF': O'F'. 
But the ratio FF' : O'F' is the ratio between the velocity of 
the earth and that of light: so that the sine of the aberration is 
equal to the ratio of the velocity of the earth to that of light, 
multiplied by the sine of the angle I. 

124. Diurnal Aberration. — Aberration causes the celestial 
bodies to appear to be nearer than they really are to that point 
of the celestial sphere towards which the motion of the earth 
is directed at the instant of observation. As a correction, then, 
it is to be applied in the opposite direction. There is evidently 
no aberration w T hen the motion of the earth is directly towards 
the star, and the greatest amount of aberration occurs when 
the direction of the earth's motion is at right angles to the 
direction of the star. 

Aberration is of two kinds, corresponding to the daily and 
the yearly motion of the earth. The diurnal aberration tends 
to displace all bodies in the direction in which the earth is 
carrying the observer : that is to say, in an easterly direction. 
It evidently varies with the linear velocity of the observer, and 
is therefore the greatest at the equator and zero at the poles. 
Owing to the minuteness of the velocity of any point of the 
earth's surface about the axis in comparison with the velocity 
of light, the diurnal aberration is extremely small, its greatest 
value being less than Jd of a second of arc. 

125. Annual Aberration. — The displacement of a star occa- 
sioned by the motion of the earth in its orbit about the sun is 
called the annual aberration. The effect which it has on the mo- 
tion of any body will depend on the relative situation of that body 



118 



ABERRATION. 



•6 


7 




<^^u~ 


A 


\ 


I 


# i 


d' 

D \ 


\~ 


J 


6' 


r \^_ 







to the plane of the ecliptic, as may be seen in Fig 50. In this 
figure S represents the sun, ABCD the orbit of the earth, and K 
the pole of the ecliptic. Suppose a star to be at K. As the earth 
moves through A, in the direction indicated by the arrow, the 
star will be displaced from iTto a; as the earth moves through 

B, the star will be seen 
at b y &c. Since the 
direction of this star is 
always at right angles 
to the direction in 
which the earth is 
moving, the aberra- 
tion will continually 
be at its maximum, 
as shown in the pre- 
vious article, and the 
star will describe a 
circle about its true 
place as a centre. If 
the star is in the plane 
of the ecliptic, as at s, there will be no aberration when the 
earth is at A or C, and the aberration will be at its maximum 
when the earth is at B or D. The star will therefore during 
the year describe the arc b'd\ equal in value to twice the maxi- 
mum of aberration, and having the true place of the star at its 
middle point. If the star is situated between the pole and the 
plane of the ecliptic, it will describe an ellipse, the semi-major 
axis of w T hich is the maximum of aberration, and the semi- 
minor axis of which increases with the latitude of the star. 

126. Velocity of Light — The maximum value of aberration is 
the same for all bodies, and may be obtained by observing the 
apparent motion of a fixed star during the year. Its value 
has thus been obtained, and is 20".4. Now, since the maximum 
of aberration occurs when the angle J, in the formula in Art. 
123, is 90°, we shall have, denoting this maximum by A\ 

Sin Jt — sy jpr 

But FF' is the velocity of the earth in its orbit, or 18.4 miles 



c 

Fig. 50. 



ABERRATION. 119 

a second. Hence, the velocity of light in a second is 18. -I 
miles multiplied by the cosecant of 20".4, which will be found 
to be 185,600 miles. Experiments of a totally different character 
have given almost precisely the same result; and it is believed 
that this estimate is within a thousand miles of the true velocity. 

If we divide the distance of the earth from the sun by this 
velocity, we find that it requires 8m. 18s. for light to pass over 
that distance. When we look at the sun, therefore, we see it, 
not as it is at the time of observation, but as it was 8m. 18s. 
previously; and in the same way every other celestial body 
appears to be in a different position from that which it really 
occupies at the instant we observe it. It may be well to notice 
here, that the time required for light to pass from any celestial 
body to the earth is an element in the computation of the appa- 
rent place of that body given in the Nautical Almanac. In 
the case of a body which changes its actual position on the celes- 
tial sphere, as a planet, for instance, allowance must also be made 
for what is called planetary aberration ; since even if the earth 
were stationary, the apparent position of the body would be 
behind its true position by the amount of its motion in the time 
required for light to come from the body to the earth.* 

127. Aberration a Proof of the Earth's Revolution about the 
Sun. — The existence of the phenomenon of aberration, as de- 
scribed in Art. 125, is a matter of undoubted observation : and 
when the close agreement of the velocity of light obtained in 
the preceding article with the velocity obtained by independent 
philosophical experiments is taken into consideration, it is fair 
to regard the existence of aberration as a strong direct proof of 
the revolution of the earth about the sun. Another proof, simi- 
lar in many respects to this, will be noticed when we come to 
the subject of the eclipses of Jupiter's satellites. 

* Herschel suggests ( Outlines of Astronomy, $ 335) that this might be 
called the equation of light, in order to prevent its being confounded with the 
real aberration of light. 



120 ORBIT OF THE MOON. 



CHAPTER IX. 

THE MOON. 

128. The Orbit of the Moon. — While the moon, in common 
with all the celestial bodies, has the apparent westward motion 
which is due to the rotation of the earth, it also changes its rela- 
tive position to the other bodies, and is continually falling behind, 
or to the east of them, in this diurnal motion. In other words, 
it has an independent motion, either real or apparent, from west 
to east. This eastward motion is so rapid, that we only need to 
observe the relative situations of the moon and some conspicuous 
star, during a few hours on any favorable night, to notice a per- 
ceptible change in their angular distance. If the right ascension 
and the declination of the moon are determined from day to day, 
precisely as the same elements of the sun's position were deter- 
mined (Art. 89), and the corresponding positions are laid down 
upon a celestial globe, we shall find that the moon makes a com- 
plete revolution in the heavens, about the earth as a centre, in 
an average period of 27d. 7h. 43m. 11.5s. We shall also find 
that the plane of the moon's orbit intersects the plane of the 
ecliptic at an angle whose mean value is 5° 8' 44", and in a line 
which, like the earth's line of equinoxes, is continually revolving 
towards the west : so that the apparent orbit of the moon is not 
a circle, but a kind of spiral. This revolution is much more 
rapid, however, in the case of the moon, the amount of retro- 
gradation being about 1° 27' in a month, and the complete revo- 
lution being effected in 18.6 years. 

The movement of the moon in its orbit is represented in Fig. 
51. Let E be the earth, and the circle MANC the plane of the 
ecliptic. Let the moon be at M at any time. Then will MN be 
the line in which the plane of the moon's orbit intersects the 
plane of the ecliptic. This line is called the line of the node*. That 



ORBIT OF THE MOON. 121 

extremity of the line through which the moon passes in moving 

from the southern to the northern v £!_-—»*•—. 

side of the ecliptic is called the s/s\ 

ascending node, the other the de- / /' \ 

scending node. Let the moon // 

move on from M in the arc MB. A 

When it descends to the ecliptic, ; \ 

it will meet it, not at the point N> \ \. . / 

but at some point N' ', and the line v - — --^jftr- J^ r 

of the nodes will take the new posi- Fig. 51. 

tion N'M'. The moon moves on to the other side of the ecliptic, 

passes through the arc N'T), and when it again returns to the 

ecliptic, will meet it, not at M\ but at some point J/", and the 

line of the nodes will take the position M"N". The revolution 

of the line of the nodes is evidently in an opposite direction to 

that in which the moon itself revolves, and is therefore from 

east to west. 

129. Cause of the Retro gradation of the Nodes. — This retro- 
grade movement of the moon's nodes is similar in character to 
the precession of the equinoxes, and is due to the attraction 
which the sun exerts upon the moon. Since the plane of the 
moon's orbit is inclined to the plane of the ecliptic, the attrac- 
tion of the sun will, in general, tend to draw the moon out of its 
orbit towards the ecliptic. The only exceptions to this rule will 
occur when the moon is at one of its nodes, and is therefore in 
the plane of the ecliptic, and also when the line of the moon's 
nodes passes through the sun, at which time the attraction of the 
sun is exerted along this line, and consequently in the plane of 
the moon's orbit. In Fig. 51 let the moon be at B, and the sun 
anywhere in the ecliptic except on the line of the nodes. As the 
moon moves on, the sun is continually drawing it down to the 
ecliptic, and it will hence meet the ecliptic, not at N, but at N f . 
The same effect will be seen at every other position of the moon 
in its orbit, with only the exceptions already mentioned. 

130. Change in the Obliquity of the Moon's Orbit. — It is evi- 
dent from the same figure that the angle w ? hich the arc BN f 
makes with the plane of the ecliptic is greater than the angle 
which an arc drawn through B and N would make. The obli- 



122 ORBIT OF THE MOON. 

quity of the plane of the moon's orbit is therefore increased a? 
the moon approaches the node. It may be shown in the same way 
that the obliquity is diminished as the moon recedes from the node. 
The extreme limits which this angle attains are 5° 20' 6" and 
4° 57' 22". 

131. Elliptical Form of the Moon's Orbit. — The angular dia- 
meter of the moon varies at different points of its orbit, while 
its mean value remains the same from month to month; w T e there- 
fore conclude, as we concluded in the case of the sun, that its 
distance from the earth is not constant, the greatest distance cor- 
responding to the least diameter, and the least distance to the 
greatest diameter. If we neglect the retrogradation of the 
moon's nodes, and represent graphically the moon's orbit by a 
method identical with the method employed in representing the 
earth's orbit (Art. 98), we shall find the orbit to be an ellipse, 
with the earth at one of the foci. The eccentricity of the ellipse 
is 0.0549, or very nearly T Vth. 

132. Line of Apsides. — That point in the moon's orbit where 
it is the nearest to the earth is called the perigee, and that point 
where it is the farthest from the earth, the apogee. The line 
connecting these two points is called the line of apsides. It is 
also the major axis of the moon's orbit. This line revolves in the 
plane of the moon's orbit from w 7 est to east, making a complete 
revolution in very nearly nine years. 

The following description of the moon's orbit, and of the 
changes to which it is subject, is given by Herschel in his Out- 
lines of Astronomy. " The best way to form a distinct conception 
of the moon's motion is to regard it as describing an ellipse about 
the earth in the focus, and at the same time to regard this ellipse 
itself to be in a twofold state of revolution ; 1st, in its ow T n 
plane, by a continual advance of its own axis in that plane; 
and 2dly, by a continual tilting motion of the plane itself, 
exactly similar to, but much more rapid than, that of the earth's 
equator." 

133. Variation in the Moon's Meridian Zenith Distance. — From 
the formula in Art. 76 we have, 

z = L — d 
At any place, then, the latitude remaining constant, the least 



DISTANCE OF THE MOON. 123 

meridian zenith distance will occur when the moun's declination 
has the same name as the latitude, and is at its maximum; and 
the greatest will occur when the declination has the opposite 
name, and is also at its maximum. Since the plane of the 
moon's orbit is inclined, at the most, 5° 20' to the plane of the 
ecliptic (Art. 130), the greatest value of the declination, either 
north or south, is 5° 20' + 23° 27'. The variation in the meri- 
dian altitude will therefore be double this amount, or 57° 34'. At 
Annapolis, in latitude 38° 59' N., the greatest altitude is 79° 48', 
the least 22° 14'. There is an exception to this general rule in 
the case of those places whose latitude is less than 28° 47': since 
at those places the greatest altitude occurs when the moon is in 
the zenith, or, as is evident from the formula, when the declina- 
tion is equal to the latitude, and has the same name. 

The new moon is in the same part of the heavens that the sun 
is in (Art. 139), and the full moon is in the opposite part. Since 
the sun attains its least altitude in winter and its greatest in 
summer, new moons will run low in winter and full moons will 
run high: while in summer the opposite of this will take place. 

DISTANCE, SIZE, AND MASS OF THE MOON. 

134. Since the sine of the moon's horizontal parallax is the 




Fig. 52. 

ratio of the radius of the earth to the distance of the moon from 
the earth, it is evident that we can determine this distance as 
soon as we obtain the horizontal parallax. The horizontal 
parallax may be found in the following manner: — -In Fig. 52, let 
be the centre of the earth, EQ its equator, and A and B the 
positions of two observers on the same meridian, w T hose zeniths 



124 DISTANCE OF TIIE MOON. 

are Zand Z '. Let if be the moon's position when crossing the 
meridian. The apparent zenith distance at A, corrected for 
refraction, is the angle ZAM, the geocentric zenith distance is 
ZOM, and the difference of these two angles, the angle A MO, 
is the parallax in altitude. In the same way OMB is the pa- 
rallax at B. Represent the parallax at A by p, that at B by 
p, the horizontal parallax by P, the apparent meridian zenith 
distance at A by z, and that at B by z\ 
We have, by Geometry, 

p = z — AOM, 
p f = z* — BOM, 
and, consequently, 

" p + p f = z + z' — AOB. 
We have also, from Art. 54, 

jo = P sin z, 
p r — P sin z\ 
and, therefore, 

p -f- p f = P (sin z + sin z'). 
Combining the two equations, and finding the expression for P, 

z + z' — AOB 
sin z -f- sin J 
But A OB is evidently the difference of latitude of A and B. 
We have, then, as our method of finding the moon's horizontal 
parallax, to subtract the difference of latitude of the two places 
from the sum of the apparent zenith distances, and to divide 
the remainder by the sum of the sines of the two zenith dis- 
tances. 

It is important that the two places of observation shall 
differ widely in latitude. It is not, however, necessary that 
they shall be on the same meridian, since, either from tables 
of the moon's motion, or from actual observation on successive 
days before and after the time of observation, we can obtain 
the change of meridian zenith distance corresponding to any 
known difference of longitude, and thus reduce the two observed 
zenith distances to the same meridian. 

135. The Moon's Horizontal Parallax. — By observations simi- 
lar to those above described, the mean value of the moon's 
equatorial horizontal parallax is found to be 57' 3". The mean 



MAGNITUDE AND MASS. 125 

distance of the moon from the earth is therefore 3962.8 miles 
multiplied by the cosecant of 57' 3", or 238,800 miles. The hori- 
zontal parallax varies between the limits of 61' 32" and 52' 50", 
and the distance between 257,900 and 221,400 miles. It must 
be noticed that by the mean value of the horizontal parallax 
given above is not meant the half sum of the two extreme 
values, but the value which the parallax has when the moon is 
at its mean distance from the earth. 

136. Magnitude of the Moon, — The angular semi-diameter of 
the moon at its mean distance from the earth is found to be 
15' 33."5. Its linear semi-diameter is therefore obtained by 
multiplying the mean distance by the sine of 15' 33". 5, and 
is found to be 1080.8 miles, or about T 3 T ths of the radius of the 
earth. The volumes of two spheres being to each other as the 
cubes of their radii, the volume of the moon will be found to 
be about ^th of that of the earth. 

137. Mass of the Moon. — The mass of the moon is obtained 
by the following considerations. If the sun does not affect the 
gravity of the moon to the earth, and the mass of the moon is 
inappreciable in comparison with that of the earth, then the 
centrifugal force of the moon in its orbit ought exactly to equal 
the attraction of the earth on the moon. But if the moon has 
a sensible mass, it will, by the law of gravitation, attract the 
earth, and its centrifugal force must be sufficient to counter- 
balance the sum of the mutual attractions of the earth and 
the moon. Now, if we refer to what was demonstrated in Art. 
113, we find that the moon's actual centrifugal force is greater, 
by -g^th, than the attraction of the earth at the distance of the 
moon. This would make the mass of the moon -g^th of that of 
the earth. But it is found that the sun diminishes sensibly the 
gravity of the moon to the earth. This, therefore, must also be 
taken into account, and the resulting value of the mass of the 
moon is found to be about -g^st of that of the earth. 

The density of the moon, being directly as the mass, and 
inversely as the volume, will be ff, or about |ths of the density 
of the earth. 

138. Augmentation of the Moon's Semi- Diameter. — If at any 
time we measure the angular semi-diameter of the moon, we 



126 



PHASES. 




shall find that it increases with the moon's altitude, being least 
when the moon is in the horizon and greatest when in the 
zenith. This increase is explained in Fig. 53. Let E be the 

centre of the earth, and 
-(J^^ ifthatofthemoon. With 

the distance between E and 
M as a radius, describe the 
semi-circumference AM'B. 
When the moon is in the 
horizon of the point C, its 
distances from C and from 
E are very nearly equal. 
Fig. 53. But as the moon rises, the 

distance CM continually decreases, while EM, the distance of the 
moon from the earth's centre, remains sensibly constant. When 
the moon is in the zenith, or at M', the distance CM' is less than 
EM ' by the radius of the earth. Now, the angular semi-diame- 
ter of the moon will increase, as shown in Art. 98, very nearly 
as the distance of the moon from the observer decreases. But 
the earth's radius is about g^th of the distance of the moon 
from the earth's centre: therefore the semi-diameter of the 
moon in the zenith will be greater than the semi-diameter in 
the horizon by ^th of itself, or by about 15". This increase 
is called the augmentation of the moon's semi-diameter. 

THE MOON'S PHASES. 

139. Two bodies are said to be in conjunction when they have 
the same longitude. They are said to be in opposition when 
their longitudes differ by 180° ; and in quadrature when their 
longitudes differ by either 90° or 270°. 

The moon is an opaque body, which is rendered visible to 
us by the rays of light which it reflects from the sun. The 
phases of the moon are due to the different relative positions 
to the sun and the earth which it has while revolving about 
the earth. 

In Fig. 54 let E be the earth, and the circle A CFH the orbit 
of the moon. Since the inclination of the plane of the moon's 
orbit to the plane of the ecliptic is only a few degrees, we may 



PHASES. 



127 



neglect it in this ease, and suppose the two planes to coincide. 
Let the sun lie in the direction ES. Since the distance of the 




Fig. 54. 

sun from the earth is about 387 times the distance of the moon 
from the earth, the lines ES, ITS, BS, &c, drawn to the sun 
from different points of the moon's orbit, may be considered to be 
sensibly parallel. Let us first suppose the moon to be in con- 
junction with the sun at the point A. Here only the dark 
portion ot the moon is turned towards the earth, and the moon 
is therefore invisible. This is called new moon. As the moon 
moves on towards B, the enlightened part begins to be visible, 
and when it reaches C, 90° in longitude from the sun, half the 
enlightened part is visible, and the moon is at its first quarter. 
When the moon is at F, in opposition to the sun, all the illumi- 
nated part is turned towards the earth, and the moon is full. 
The moon wanes after leaving F, passes through its last quarter 
at IT, and finally becomes again invisible. 

Between A and C the moon is crescent, as represented at L, 
and between C and F it is gibbous, as represented at N. The 
same terms are also applied to the appearance of the moon be- 
tween H and A and between F and H. 

140. Phases of the Earth to the Moon. — It is evident from Fig. 
54 that the earth presents phases to the moon identical in cha- 
racter with those presented by the moon to the earth, although 



128 



SIDEREAL AND SYNODICAL PERIODS. 



similar phases are not presented by each body at the same time. 
Thus at the time of new moon the earth is full to the moon : 
and the light which it then reflects to the moon renders the 
unenlightened part of the moon faintly visible to the earth. As 
the moon moves on to its first quarter, the earth reflects less and 
less light to it, until finally the unenlightened portion disappears. 

SIDEREAL AND SYNODICAL PERIODS. 

141. The sidereal period of the moon is the interval of time 
in which it makes one complete revolution in its orbit about 
the earth. The sy nodical period, (or lunation) is the interval 
between two successive conjunctions or oppositions. Owing to 
the earth's revolving about the sun, and carrying the moon with 
it, the synodical period is longer than the sidereal period, as may 
be seen in Fig. 55. 

Let S be the sun, E the 
earth, and MANB the or- 
bit of the moon. Let the 
moon be at M, in conjunc- 
tion with the sun. As the 
moon moves about E in 
the curve MANB, the 
earth also moves about the 
sun in the direction EE f . 
The next conjunction will 
therefore not occur until the 
moon reaches M". Now, 
rig. 55. if through E r we draw the 

line M'N' parallel to MN, the sidereal period of the moon is 
completed when the moon reaches M' . The synodical period is 
therefore greater than the sidereal period by the time required 
by the moon to pass through the angle M'E'M". This angle is 
evidently equal to the angle ESE f , which is the angular advance 
of the earth in its orbit in the period of one synodical revo^ 
lution of the moon. In one lunar month, then, the angular ad- 
vance of the moon in its orbit is greater by 360° than the angu- 
lar advance of the earth in its orbit. If, therefore, we denote 
the moon's sidereal period in days by P, its synodical period by 




SIDEREAL AND SYNODICAL PERIODS. 129 

8, and the earth's sidereal period, or one sidereal year, by T, we 
shall have, 

— — = the earth's daily angular velocity, 

360° 

— - p— = the moon's daily angular velocity, 

360° 

— — = the moon's daily angular gain on the earth. 

Hence we shall have, 

360° 360° 360°. 



P = 



T 
ST 



S+T 

The sidereal period of the moon is therefore obtained by mul- 
tiplying the sidereal year by the moon's synodical period, and 
dividing the product by the sum of the sidereal year and the 
synodical period. 

142. Values of the Synodical and Sidereal Periods.- — The value 
of the synodical period is not constant, but varies from month 
to month. A mean value may, however, be obtained by divid- 
ing the interval of time between two oppositions, not conse- 
cutive, by the number of synodical revolutions in that interval. 
Now, the day, the hour, and even the probable minute, at which 
an opposition of the moon occurred in the year 720 B.C., were 
recorded by the Chaldeans; and by comparing this time with 
the results of recent observations, an extremely accurate value 
of the mean synodical period is obtained It is found to be 
29d. 12h. 44m. 3s. We have,- then, for the value of the side- 
real period, by the formula in the preceding article, 

_ 3 65.256 X 29.53 
~ 365.256 -f- 29.53 * : 
whence we obtain the value already given in Art. 128. 

143. Retardation of the Moon, and the Harvest Moon. — The 
mean daily motion of the moon towards the east is about 13°, 
while that of the sun is, as we have already seen, about 1°: 
hence the moon is continually falling to the rear of the sun in 
apparent westward motion, and the interval of time between any 
two successive transits of the moon is greater than the similar 

y 



130 ROTATION OF THE MOON. 

interval in the case of the sun. The moon, therefore, rises later, 
and sets later, day by day. This is called the retardation of the 
moon. Its amount varies considerably in value, but is on the 
average about fifty minutes. 

The less the angle which the plane of the moon's orbit makes 
with the plane of the horizon, the less does the advance of the 
moon carry it with reference to the horizon, and, consequently, 
the less is the retardation of the moon in rising. Now 7 , since 
the moon's orbit very nearly coincides with the ecliptic, the 
retardation in rising will in general be the least, when the 
ecliptic makes the least angle with the horizon. By reference 
to a celestial globe, it will be seen that the ecliptic makes the 
least angle with the horizon when the vernal equinox is in the 
eastern horizon. The least retardation in rising, therefore, 
occurs in each month when the moon is near the sign of Aries. 
This least retardation is especially noticeable when it occurs at 
the time of full moon. Now, when the moon is in Aries, and 
full f the sun must be in Libra, or near the autumnal equinox. 
This occurs about the 21st of September. About the time, then, 
of the full moon which occurs near the 21st of September, the 
moon will rise, for two or three nights, only about half an hour 
later each night. Usually this small retardation is noticed at 
the times of two full moons, one in September and the other in 
October. The first is called the Harvest Moon, the second the 
Hunter's Moon. All this relates to the Northern Hemisphere. 

ROTATION. LIBRATIONS AND OTHER PERTURBATIONS. 

144. Rotation of the Moon. — By observation of the spots upon 
the disc of the moon, it is found that very nearly the same 
surface of the moon is turned continually towards the earth. 
The conclusion draw r n from this fact is that the moon rotates 
upon an axis in tfcp same time in which it revolves about the 
earth, or in 27.3 days. The plane in which this rotation is 
performed makesan angle of about 1° 32' with the plane of the 
ecliptic. 

If there are any inhabitants of the moon, their day will be 
equal in length to about twenty-seven of our days, and their 
night to about twenty-seven of our nights. Since the plane of 



L1BKATIOXS OF THE MOON. 



131 



the moon's equator is so nearly coincident with the plane of this 
ecliptic, there will hardly be any sensible change of seasons : 
or if there is, the lunar day will be the lunar summer, and the 
night the winter. To the inhabitants of one hemisphere the 
earth will be perpetually invisible, while to the inhabitants of 
the other hemisphere it will present the appearance of a body 
very nearly stationary in their sky, exhibiting phases similar 
to those which we see in the moon, with a radius nearly four 
times that of the moon, and a surface about thirteen times that 
of the moon. 

145. Librations. — By libration is meant an apparent oscilla- 
tory movement of the moon, which enables us, in the course of 
its revolution, to see something more than an exact hemisphere. 

The libration in longitude is due to the fact that the moon's 
rotation on its axis is perfectly uniform, while its motion about 
the earth is not. Hence the line drawn from the centre of the 




l 

Pig. 56, 

earth to that of the moon does not always intersect the surface 
of the moon at the .same point, and we are able at times to look 



132 OTHER PERTURBATIONS. 

a few degrees, east or west, b?yond the mean visible border. If, 
in Fig. 56, ABCD represents the earth, E its centre, and R 
the centre of the moon, the dotted lines at N denote the limits 
between which, as the moon revolves about the earth, the visible 
border may deviate from its mean position. 

The Vibration in latitude is due to the fact that the axis of the 
moon, remaining constantly parallel to itself, is not perpen- 
dicular to the plane of the moon's orbit, but is inclined to it at an 
angle of about 83° 19'. We are therefore able at certain times 
to see about 6° 41' beyond the north pole of the moon, and at 
other times the same amount beyond the south pole. Thus in 
Fig 56, when the moon is at M y we can see beyond the pole P, 
and when the moon is at 0, beyond the pole p: since in each 
case we can see nearly that portion of the moon which lies be- 
tween the earth and the circle ab, whose plane is perpendicular 
to the plane of the moon's orbit. 

The diurnal Vibration is due to the difference between that 
hemisphere of the moon which is turned towards the centre of 
the earth and that which is turned towards any point on the 
surface. When, for instance, the moon is at Z, an observer at 
C will see the same hemisphere which is turned towards the 
earth's centre, while an observer at G will see a different one. 
The hemisphere which is turned towards any observer when 
the moon is rising will also be different from the one which is 
turned towards him when the moon is setting. It is evident in 
the figure that the amount of this libration varies with the angle 
ERG; that is to say, with the moon's parallax. 

Notwithstanding all these librations, we are able to see in all 
only about T 5 o 7 Q tns °f the moon's surface, according to Arago: 
the remainder being continually concealed from our view. 

146. Other Perturbations. — Besides these librations, and the 
perturbations already mentioned (Art. 132), there are other per- 
turbations in the moon's motion of which only a very brief notice 
can here be given. The greatest of these perturbations is called 
evection, and was discovered by Ptolemy in the second century. 
It arises from the variation in the eccentricity of the moon's 
orbit, and from the fluctuations in the general advance of the 
line of the apsides. By it the moon's mean longitude is alternately 



LUNAR CYCLE. 133 

increased and decreased by about 1° 20'. Another perturbation 
in the moon's motion is called variation. It depends solely on 
the angular distance of the moon from the sun, and its maximum 
is 37'. The annual equation depends on the variable distance 
of the earth from the sun, and amounts to 11'. The secular acce- 
leration is an increase in the moon's motion which has been going 
on for many centuries, at the rate of about 10" a century. This 
perturbation is the result of the diminution of the eccentricity 
of the earth's orbit; and from what has been said on that subject 
in Art. 98, it is evident that this inequality will at some distaut 
day become a secular retardation. 

All of these perturbations are satisfactorily explained by the 
investigation of what is known as the problem of the three bodies, 
in which two bodies are supposed to revolve about their common 
centre of gravity, according to the law of universal gravitation, 
and the effects of the attraction exerted by a third body upon 
the motions of these two bodies are made the object of mathe- 
matical examination. 

THE LUNAR CYCLE. 

147. If we multiply the number of days, hours, &c, in a 
synodical period of the moon (Art. 142) by 235, the product 
will be 6939d. 16h. 27m. 50s. Now, in a period of nineteen civil 
years there are either 6939 days, or 6940 days, according as there 
are four or five leap years in that period. If, then, in any year, 
new moon occurs on any particular day of the month, the first 
of January, for instance, it will occur again on the first of Janu- 
ary (or at all events within a few hours of its end or beginning), 
after an interval of nineteen years ; and all the new moons and 
the other phases will occur on very nearly the same days through- 
out the second period of nineteen years on which they occurred 
during the first period. This period is called the Lunar Cycle. 
It is also called the Metonic Cycle, having been originally dis- 
covered, B.C. 432, by Meton, an Athenian mathematician. The 
present lunar cycle began in 1862. 

This cycle is used in finding Easter: Easter being the first 
Sunday after the full moon which occurs either upon or next 
after the 21st dav of March. 



134 APPEARANCE OF THE MOON. 

The golden number of any year is the number which marks 
the place of that year in the cycle. It may be found for any 
year by adding 1 to the number of that year, and dividing the 
sum by 19; the remainder (or 19, if there is no remainder) is 
the golden number. 

Four lunar cycles, or seventy-six civil years, constitute what 
is called the Callippic cycle. 

GENERAL DESCRIPTION OF THE MOON. 

148. When viewed through powerful telescopes, the surface 
of the moon is found to be made up of mountains, valleys, and 
plains, similar in general appearance to those that exist on the 
earth. As a whole, however, the surface of the moon is much 
more uneven than that of the earth. The heights of over 1000 
lunar mountains have been measured, and some of them have 
been found to exceed 20,000 feet. Many of these mountains 
bear the appearance of having been at one time volcanoes, far 
surpassing in size and activity those on the earth. The common 
belief among astronomers has been that these lunar volcanoes 
are now extinct. Messrs. Beer and Madler, two Prussian astro- 
nomers who have made the moon their special study, have de- 
tected no signs of activity in any of the volcanoes which they have 
examined. Recently, however, certain phenomena have been 
noticed which seem to show that one at least of these volcanoes, 
named Linne, is not extinct: but the character of the pheno- 
mena is not thus far sufficiently decisive to settle the question. 

There are no signs of the existence of water on the moon. 
Certain large dark patches are seen, which were formerly con- 
sidered to be oceans, gulfs, &c, and were so named ; but increased 
telescopic power shows that they are dry plains. 

It seems to be still an open question whether or not the moon 
has an atmosphere. If there is an atmosphere, it must be of an 
extremely minute height and density ; for we see no clouds and 
no twilight, and there is nothing in the phenomena of the occul- 
tations of stars by the moon which shows the existence of even 
the rarest atmosphere. Some observers, however, and among 
them Messrs. Beer and Madler, believe that they have detected 
signs of the existence of a very slight atmosphere. 



PLATE II 




TOTAL ECLIPSE OF THE SUN, OF JULY 18, 1860, 

showing the Corona and the Red Flames; as observed by Dr. 
Feilitzsch, at Castellon de la Plana, 



LUNAR ECLIPSE. 135 



CHAPTER X. 

LUNAR AND SOLAR ECLIPSES. OCCULTATIONS. 

149. Eclipses. — The obscuration, either partial or total, of the 
light of one celestial body by another is in astronomy termed an 
eclipse. When the earth comes between the sun and the moon, 
the light of the sun is shut off from the moon, and we have a 
lunar eclipse. A lunar eclipse can occur only at the time when 
the moon is in opposition to the sun, that is to say, at the time 
of full moon. When the moon comes between the earth and the 
sun, the light of the sun is shut off from the earth, and we have 
a solar eclipse. A solar eclipse can occur only at the time of 
new moon. An eclipse of a star or a planet by the moon is called 
an occultation. 

If the orbit of the moon lay in the plane of the ecliptic, a 
lunar and a solar eclipse would occur in every month. Owing, 
however, to the inclination of the plane of the moon's orbit to 
the plane of the ecliptic, the latitude of the moon is usually too 
great to allow either kind of eclipse to take place; and it is only 
in special cases, when the moon is in or near the plane of the 
ecliptic at the time of conjunction or opposition, that an eclipse 
of the sun or the moon is possible. 

LUNAR ECLIPSE. 

150. In Fig. 57 let S be the centre of the sun, and E that of 

B M' 




Fig. 57. 



136 LUNAR ECLIPSE. 

the earth. Draw the lines BH and CG, tangent to the two 
spheres. These lines will meet at some point A, and AHEG 
will be a section of the shadow cast by the earth. 

The whole shadow is of a conical shape, the vertex of the cone 
being at A ; and a lunar eclipse will occur whenever the moon is 
within this shadow. Draw 7 the tangent lines BG and CH. 
KDL is a section of a second cone whose vertex is at D. The 
earth's shadow is called the umbra, and that portion of the second 
cone which lies outside of the umbra is called the penumbra. 
Thus KHA and A GL are sections of the penumbra. It must 
be noticed, in regard to the construction of this figure, that since 
only one tangent can be drawn to the circumference of a circle 
at any one point, the lines BG and CH do not touch the two 
circumferences at precisely the same points at which BH and 
CG touch; the points of tangency are, however, so nearly iden- 
tical as to be indistinguishable in the figure. 

Now let M'MM" represent a portion of the moon's orbit at the 
time of a lunar eclipse. As soon as the moon passes within the 
line DK, some of the rays of the sun will be cut off from it by 
the earth, and its brightness will begin to decrease. The whole 
disc, however, will still be visible. As soon as the moon begins 
to pass within the line HA, the disc will begin to disappear, and 
when the whole disc has passed within the cone, the eclipse will 
be total. 

151. Different Kinds of Eclipses. — When the moon's orbit 
is so situated that only a part of the moon enters the umbra, 
we have a partial eclipse. When the moon does not enter the 
umbra, but merely touches it, we have an appulse. When the 
centre of the moon coincides with the line which connects the 
centre of the earth and that of the sun, the eclipse is cen- 
tral. A central eclipse occurs very rarely, if indeed it occurs 
at all. 

152. The Semi-Angle of the Umbral Cone. — The semi-angle of 
the umbral cone is the angle EA G, Fig. 58. Now we have, by 
Geometry, 

SEC=ECG + EAG. 
But SEC is the sun's angular semi-diameter, and ECG is its 



LUNAR ECLIPSE. 



137 



horizontal parallax. Putting S for SEC, and P for ECG, we 
have, 

EAG = S— P. 




153. The Angular Semi-Diameter of the Shadow at the Distance 
of the Moon. — The angular semi-diameter of the shadow at the 
distance of the moon is the angle MEM'. We have, by Geometry, 

EM ' G = MEM' + EA G. 
Now EM' G is the moon's horizontal parallax, which we will 
represent by P', and the value of EAG has been obtained in 
the preceding article. We therefore have, 

MEM' = P'+P — S. 

Observation shows that the earth's atmosphere increases the 
apparent breadth of the shadow by about its one-fiftieth part : 
hence in practice the angular semi-diameter of the shadow is 
taken equal to § J (P* -\- P — S ). If we substitute in this expres- 
sion the least values of P f and P, and the greatest value of S, 
from the table given in Art. 155, we shall find that the least 
value of the angular semi-diameter of the shadow is about 37' 25" : 
so that the entire breadth of the shadow is always more than 
double the greatest diameter of the moon. 

154. Length of the Earth's Shadow. — The length of the shadow, 
or the line EA, can be computed from the right-angled triangle 
EA G, in which we have, 

EA=EGcosec(S — P). 

The mean value of this length is 858,000 miles, or more than 
three times the distance of the moon from the earth. 

155. Lunar Ecliptic Limits. — We see from Fig. 5S that a lunar 
eclipse can occur only when the moon's geocentric latitude at 
the time of opposition, (or at full moon,) is less than the sum 
of the angular semi-diameter of the shadow and the semi-diameter 



138 



LUNAR ECLIPSE. 



of the moon. If we represent the moon's semi-diameter by 5\ 
the expression for this sum is 

51 



50 



(P + -F— S)+&. 



If the moon's geocentric latitude at the time of opposition is 
greater than the greatest value which this expression can attain, 
no eclipse can possibly occur : if it is less than the least value of 
the expression, an eclipse is inevitable. These two values of 
this expression are called the lunar ecliptic limits. Now, w r e have 
by observation the following values of P, P', &c. : 



P' 


MAXIMA. 


MINIMA. 


61' 32" 


52 / 50" 


p 


9 


9 


s / 


16 46 


14 24 


s 


16 18 


15 45 



In order to find the greatest value of the expression, we sub- 
stitute in it the greatest values of P, P and $', and the least value 
of S. The result is 1° 3' 37" : and no eclipse will occur when the 
moon's latitude exceeds this limit. The least value of the expres- 
sion is 51' 49": and when the moon's latitude at opposition is less 
than this, an eclipse cannot fail to occur. There are some con- 
siderations, however, which have not been taken into account, 
which may increase each of these limits by about 16". 

When the moon's latitude at opposition is within these limits, 
an eclipse is possible, but not necessarily certain. In order to 
determine whether in such case it will or will not occur, the 
actual values which P, P', S and S' will have at that time must 
be substituted in the expression, and the result compared with 
the corresponding latitude of the moon. 

156. Since a lunar eclipse is caused by the moon's entering 
the earth's shadow, it will be seen at the same instant of time by 
every observer who has the moon above his horizon: and the 
character of the eclipse, whether total or partial, will be every- 



SOLAS ECLIPSE. 139 

where the same. As the moon's motion towards the east is more 
rapid than that of the earth (and consequently of the shadow), 
the eclipse will begin at the eastern limb of the moon. A total 
eclipse of the moon may last for nearly two hours. Even when 
totally eclipsed, however, the moon does not, in general, disappear 
from view, but shines with a dull reddish light. This phenomenon 
is caused by the earth's atmosphere, which refracts the rays of 
light from the sun w r hich enter it near the points G and H, Fig. 
57, and turns them into the cone. The rays which pass still 
nearer to these points are probably absorbed by the atmosphere, 
thus giving rise to the observed increase of the shadow mentioned 
in Art. 153. 

SOLAR ECLIPSE. 

157. In Fig. 59, let S represent the sun, E the earth, and M 
the moon, at the time of a solar eclipse. HAK will be a section 
of the moon's umbra, and GHA and AKD, sections of its pen- 
umbra. 

G 




Fig. 59. 



To an observer situated w r ithin the umbra, at any point of the 
arc ab, the eclipse will be total ; while to one situated within the 
penumbra, as at L, for instance, the eclipse will be partial. 
Beyond the penumbra no eclipse whatever will be seen. Hence 
the geographical position of the observer determines the charac- 
ter of the eclipse : a condition different from that in the case of 
a lunar eclipse, which w T e have seen is the same to all observers. 

158. Length of the Moons Shadow. — It is evident that, to an 
observer at the apex of the shadow A, the angular semi-diameters 
of the sun and the moon would be equal. Now, the mean angular 
semi-diameters of these two bodies, as seen from the earth's centre, 



140 SOLAR ECLIPSE. 

are nearly equal ; hence the mean position of the apex does not fall 
very far from the earth's centre J?. An approximate value of the 
length of the shadow may be thus obtained. We have in Fig. 59, 

sin HAM= -nrp. 
AM 

But we have just now seen that HAM is the sun's angular 
semi-diameter, as seen from A ; and as AE is small compared 
with AS, we may consider the angle HAM to be the sun's geo- 
centric semi-diameter. Denoting this by S, we have, 

. ■ HM ' 

sin o = ~nrF- 
AM 

Now, if S' represents the moon's geocentric semi-diameter, we 

have, 

• o, HM 

smS =EM' 
Combining these two equations, and finding the value of AM, 
We have, 

AM=££ EM. 

sin /S 

Knowing, then, the distance of the moon from the earth's centre, 
and the semi-diameters of the sun and the moon, we may find 
the length of AM. When the two semi-diameters are equal, we 
have A M equal to EM, and the apex is at the earth's centre. 
When the semi-diameter of the moon is greater than that of 
the sun, the apex falls beyond the earth's centre: when it is 
less, the apex does not reach the centre. Appropriate calcula- 
tions will show that when both sun and moon are at their mean 
distances from us, the apex falls short of the earth's surface : 
and that when the moon is at its least distance from the earth, 
and its shadow is the longest, the apex falls about 14,000 miles 
beyond the earth's centre. 

159. Different Kinds of Eclipses. — When the shadow falls 
beyond the earth's surface, the eclipse is total, as we have 
already seen, within the umbra, and partial within the penumbra. 
When the apex just touches the earth, the eclipse is total only 
at the point where it touches. When the apex falls short of 
the surface, there will be no total eclipse ; but at the point in 
which the axis of the cone, prolonged, meets the earth, the 



*OLAK ECLIPSE. 141 

observer will see what is called an annular eclipse, the moon 
being projected upon the disc of the sun, but not covering it. 

160. Solar Ecliptic Limits. — In Fig. 60 let 8 represent the 
sun, E the earth, and M the moon. No eclipse of the sun can 
occur unless some part of the moon passes within the lines BC 

B 




G 

Fig. 60. 

and GD, drawn tangent to the sun and the earth : that is, unless 
the moon's geocentric latitude is less, than the angle MES. 
Now we have, 

MES = MEA + AEB + BES f 
and, also, 

AEB= CAE— CBE. 

BES is the sun's semi-diameter, 31EA that of the moon, CAE 

the moon's horizontal parallax, and CBE the sun's : hence, 

using the notation already employed in Art. 155, we have, 

MES = 8 + 8' + F — P. 

The greatest value of this expression is found by employing 
the greatest values of 8, 8', and P', and the least value of P, 
as given in Art. 155, and is 1° 34' 27" : and there will be no 
eclipse if the moon's latitude at conjunction is greater than 
this amount. The least value is 1° 22' 50" ; and if the latitude 
at conjunction is less than this an eclipse is inevitable. These 
two values, which, owing to certain considerations omitted in 
this discussion, should both be increased by about 25'', are called 
the solar ecliptic limits. In order to determine whether an 
eclipse will occur when the moon's latitude at conjunction falls 
within these limits, we must substitute in the expression the 
values which the different quantities will really have at that 
time, and compare the result w T ith the corresponding latitude 
of the moon. 

161. General Phenomena. — Since the moon moves towards 



142 CYCLE OF ECLIPSES. 

the east more rapidly than the sun, a solar eclipse will begin 
at the western side of the sun. For the same reason the moon's 
shadow will cross the earth from west to east, and the eclipse 
will begin earlier at the western portions of the earth's surface 
than at the eastern. The moon's penumbra is tangent to the 
earth's surface at the beginning and the end of the eclipse, so 
that the sun will be rising at that place where the eclipse is 
first seen, and setting at the place where it is last seen. A solar 
eclipse may last at the equator about 4 \ hours, and in these 
latitudes about Z\ hours. That portion of the eclipse, however, 
in which the sun is wholly concealed can only last about eight 
minutes: and in these latitudes, only about six minutes. 

The darkness during a total eclipse, though subject to some 
variation, is scarcely so intense as might be expected. The sky 
often assumes a dusky, livid color, and terrestrial objects are 
similarly affected. The brighter planets and some of the stars 
of the first magnitude generally become visible ; and sometimes 
stars of the second magnitude are seen. The corona and the 
rose-colored protuberances described in Art. 102 also make 
their appearance. When the sun's disc has been reduced to a 
narrow crescent, it sometimes appears as a succession of bright 
jDoints, separated by dark spaces. This phenomenon bears the 
name of Baily's beads. The dark spaces are supposed to be 
the lunar mountains, projected upon the sun's disc, and allowing 
the disc to show between them. 

Occasionally the moon's disc is faintly seen, shining with a 
dusky light. This is caused by the rays of the sun, reflected 
back to the moon by that portion of the earth's surface which 
is still illuminated by the sun : just as at the time of new moon 
its entire disc is rendered visible. 

CYCLE AND NUMBER OF ECLIPSES. 

162. Cycle of Eclipses. — In order that either a solar or a 
lunar eclipse shall occur, it is necessary, as we have seen, that 
the moon shall be near the ecliptic (in other words, near the 
line of nodes of its orbit), at either conjunction or opposition. 
It is evident that when the moon is near the line of nodes at 
guch a time, the sun also must be near the same line. The 



NUMBER OF ECLIPSES. 113 

occurrence of eclipses, then, depends on the relative situations 
of the sun, the moon, and the moon's nodes, and is only pos- 
sible when they are all in, or nearly in, the same straight line. 
We have already seen (Art. 128) that the line of nodes is con- 
tinually revolving to the west, completing a revolution in about 
18.6 years. The sun, then, in its apparent path in the ecliptic 
will move from one of the moon's nodes to the same node again 
in less than a year. This interval of time may be called the 
synodical period of the node, and is found to be 346.62 days. 

Now, w T e have, 

19 X 346.62d. = 6585.8d: 
and, the lunar month being 29.53 days, we have also, 
223 X 29.53d. = 6585.2d. 

If, then, the moon is full and at its node on any day, it will 
again be full, and at the same node, or very nearly at it, after 
an interval of 6585 days : and the eclipses w r hich have occurred 
in that interval will occur again in very nearly the same order. 
This period of 6585 days, or 18 years and 10 days, is called the 
cycle of eclipses. It w T as known to the Chaldsean astronomers 
under the name of Saros. Care must be taken not to confound 
this cycle with the lunar cycle described in Art. 147. 

163. Number of Eclipses. — Since the limit of the moon's lati- 
tude is greater in the case of a solar eclipse than in the case 
of a lunar eclipse, there are more solar eclipses than lunar 
eclipses. Usually 70 eclipses occur in a cycle, of which 41 
are solar and 29 are lunar. Since we know that a solar eclipse 
is inevitable when the moon is so near the line of nodes at 
conjunction that its latitude is less than 1° 23' 15", we can com- 
pute the corresponding angular distance of the sun at the same 
time from this line ; and having computed this, we may also 
determine the length of time required by the sun in passing 
through double this angle, or, in other w T ords, the time required 
in passing from one of these limits to the corresponding limit on 
the other side of the same node. If w 7 e do this, we shall find 
that the sun cannot pass either node of the moon's orbit without 
being eclipsed : and therefore there must be at least two solar 
eclipses in a year. The greatest number that can occur is five. 
The greatest number of lunar eclipses in the year is three, and 



144 OCCULTATIONS. 

there may be none at all. The greatest number of both kinds 
of eclipses in a year is seven ; the usual number is four. 

Although the annual number of solar eclipses throughout the 
whole earth is the greater, yet at any one place more lunar eclipses 
are visible than solar. The reason of this is that a lunar eclipse, 
when it does occur, is visible over an entire hemisphere, while 
the area within which a solar eclipse is visible is very much 
more limited. 

OCCTJLTATIONS. 

164. An occultation of a planet or a star will occur whenever 
the planet or star is so situated in latitude as to allow the moon 
to come in between it and the earth. In order to determine the 
limit of a planet's latitude within which an occultation of the 
planet is possible, let us refer to Fig. 61. In this figure, E is the 




Fig. 61. 

centre of the earth, P that of a planet, and M that of the moon. 
An occultation will occur when the moon comes between the 
tangent lines GB and AH. Let EC be the plane of the ecliptic. 
PEC is then the geocentric latitude of the planet, and MEC 
that of the moon. 

We have, 

PEC = PEG + GED + DEM+ MEC, 
and also, 

GED = EDB — EGD. 

Now, PEG is the planet's semi-diameter, EGD its horizontal 
parallax, DEM the moon's semi-diameter, EDB its horizontal 
parallax, and MEC, as above stated, its latitude. The value 



OCCULT 4TIONS. 145 

of PEC, therefore, can very readily be obtained. If P, instead 
of representing a planet, represents a star, the distance PE 
becomes so great that AH and BG are sensibly parallel, and 
the star's parallax and semi-diameter reduce to zero. In this 
case the greatest value of PEC, within which an occultation 
can occur, will be the sum of 5° 20' 6", 61' 32", and 16' 45", 
which is 6 D 38' 23". 

Since the moon moves from west to east, the occultation 
always takes place at its eastern limb. From new moon to full, 
the dark portion of the moon is to the east, as may be seen in 
Fig. 54, and from full moon to new, the bright limb is to the 
east. When an occultation occurs at the dark edge, particu- 
larly if the moon is so far on towards its first quarter that the 
dark portion is invisible, the disappearance is extremely strik- 
ing, as the occulted body appears to be extinguished without 
any visible interference. 

As already stated in Art. 83, a solar eclipse, or an occultation 
of a planet or star, although not visible at different places at 
the same absolute instant of time, may still be made the means 
of very accurately determining the longitude of a place, or the 
difference of "longitude of two places. For instance, in the case 
of a solar eclipse, we may deduce, from the local times of the 
beginning and the end of the eclipse, as observed at any place, 
the time of true conjunction of the sun and the moon: the time 
of conjunction, that is to say, as seen from the centre of the earth. 
If, then, w T e compare the local time of true conjunction with the 
Greenwich time of true conjunction, it amounts to comparing 
the local and the Greenwich time corresponding to the same 
absolute instant : and the difference of these two times will evi- 
dently be the longitude of the place of observation from Green- 
wich. 



10 



146 



TIDES. 



CHAPTER XI. 



THE TIDES. 

165. The surface of the ocean rises and falls twice in the 
course of a lunar day, the length of which is, as we have already 
seen (Art. 143), about 24h. 50m. of mean solar time. The rise 
of the water is called flood tide, and the fall ebb tide. When the 
water is at its greatest height it is said to be high water, and 
when at its least height, low water. 

166. Cause of the Tides. — The tides are due to the inequality 
of the attractions exerted by the moon upon the earth and the 
waters of the ocean, and to a similar but smaller inequality in 
the attractions exerted by the sun. 

In order to examine the phenomena of the tides, we will con- 
sider the earth to be a solid globe, surrounded by a shell of 
water of uniform depth. The centrifugal force induced by the 
rotation of the earth would tend to give a spheroidal form to 
this shell of w T ater; but as we wish simply to examine the effects 
of the attractions exerted by the moon and the sun, w 7 e will dis- 
regard the rotation of the earth, and consider it to be at rest. 

In Fig. 62, then, let AB CD represent the earth, and the dotted 




oif 



Fig. 62. 



line GHIK the surrounding shell of water. Let M be the moon. 
The attraction of the moon on the solid mass of the earth is the 



TIDES. 147 

&ame that it would be if the whole mass were concentrated at 
the point E. Now since, by the law of gravitation, the at- 
traction of the moon on any two particles is inversely as the 
square of the distances of the two particles from the moon, the 
attraction exerted upon the particle of water at Gwill be greater 
than that exerted upon the general mass of the earth, supposed 
to be concentrated at E. The particle G will therefore tend to 
recede from the earth: that is to say, its gravity towards the 
earth's centre will be diminished, although, as is plain, it will not 
move. The same result will follow at the opposite point L 
The moon will exert a greater attractive power upon the mass 
of the earth than upon a particle at J, and will tend to draw the 
earth away from the particle: so that the gravity of the particle 
at /towards the earth's centre will also be diminished. Since 
the ratio of the distances ME and MG is very nearly equal to 
the ratio of the distances MI and ME, the amount of the decrease 
of gravity at G and at J will be nearly the same. 

Let us next examine the effect of the moon's attraction at 
some point L, not situated vertically under the moon. The at- 
traction of the moon at this point is less than that at the point 
G, since the distance ML is greater than the distance MG; and 
since the attraction exerted on the mass of the earth is, of course, 
the same for both points, the difference of the attractions exerted 
on the earth and the water is less at the point L than at the 
point G. At the point L, however, this inequality of attraction 
is not wholly counteracted by gravity : for if the force with which 
the moon tends to draw a particle at L along the line ML be re- 
solved into two forces, one in the direction of the radius EL> 
and the other in the direction of the tangent XT, the latter force 
will cause the particle to move towards the point G. The same 
result will follow at any other point of the arc HGK: so that 
all the water in that arc tends to flow towards the point G, and 
to produce high water there. 

In the same way it may be shown that the w T ater in the arc 
HIK tends to flow towards the point J, and to produce high 
water at that point. 

The result, then, of the attraction of the moon, exerted under 
the suppositions which we made at the outset, is to give to the shell 



148 



TIL>ES. 



of water a spheroidal form, as shown in the figure, the major 
axis of the spheroid being directed towards the moon. Suitable 
investigation shows that the difference of the major and the minor 
serni-axis of this spheroid is about fifty-eight inches. 

167. Daily Inequality of the Tides. — The rotation of the earth, 
and the inclination of the plane of the moon's Orbit to the plane 
of the equator, produce in general an inequality in the two daily 
tides at any place. In order to show this, we will suppose that 
the spheroidal form of the w r ater is assumed instantaneously in 
each new 7 position of the earth as it rotates. In Fig. 63, let E 

m be the centre of the 
earth, surrounded, 
as in Fig. 62, by 
a spheroidal shell 
of water, the trans- 
verse axis of the 
spheroid lying in 
the direction of the 
moon M. Let Pp 
be the axis of rota- 
tion of the earth, 
and CD the equator. 
The angle MED is 
the moon's declination. The water is at its greatest height, as be- 
fore, at the points A and B, and the height at other points dimi- 
nishes as the angular distance of those points from the line GH 
increases. Let J be a place having the same latitude that A has, 
but situated 180° from it in longitude. The height of the tide 
at J is represented by IK. In a little more than tw r elve hours 
the rotation of the earth will have caused I and A to change 
places with reference to the moon. J wall then be where A is 
in the figure, and will have a tide with the height A G, w T hile 
A will be where J is now, and will have a tide with the height 
IK. It is not necessary to prove that IK is less than A G. We 
see, then, that at both A and I the two daily tides are unequal, 
the greater of the two occurring at each place at the time of the 
moon's upper culmination at that place, or being, at all events, 
the one which occurs next after that culmination. The same 




TIDES. l4*J 

daily inequality of tides may be shown to exist at any other 
points on the earth's surface, as, for instance, at L and 0. At 
the equator, however, and also at the poles, the two daily tides 
are sensibly equal, as may readily be seen from the figure. 

168. General Laws. — As far as the influence of the moon is 
concerned in causing tides, the following general laws may be 
deduced from what has been shown in the preceding articles. 

(1.) When the moon is in the plane of the celestial equator, or, 
in Fig. 63, when EM coincides with ED, the tides are greatest 
at the equator, and diminish at other places as the latitude in- 
creases; and the two daily tides at any place are sensibly equal. 

(2.) When the moon is not in the plane of the celestial equa- 
tor, the two daily tides at any place except the poles and the 
equator are unequal. The greatest tides, and the greatest ine- 
quality of tides, occur at those places whose latitude is numeri- 
cally equal to the moon's declination. If the place is on the 
same side of the equator as the moon, the greater of the two 
daily tides occurs at or next after the upper culmination of the 
moon; if the place is on the opposite side of the equator, the 
greater tide occurs at or next after the lower culmination of the 
moon. 

(3.) Owing to the retardation of the moon (Art. 143), there is 
a similar retardation in the occurrence of high water. The 
length of the lunar day being on the average 24h. 50m., the aver- 
age interval of time between two successive tides is 12h. 25m. 

169. Influence of the Sun in Causing Tides, — All that has been 
said in the preceding articles with regard to the influence of the 
moon in creating tides is equally true with regard to the influ- 
ence of the sun. The mass of the sun being so immense in 
comparison wdth that of the moon, it might be supposed that 
the influence of the sun over the tides would be greater than 
that of the moon, even although its distance from the earth is 
much greater than that of the moon. But such is not the case 
in fact. The height of the tide produced by either body is not 
so much due to the absolute attraction which that body exerts, 
as to the relative attractions which it exerts on the solid mass of 
the earth and on the water: and the moon is so much nearer to 
the earth than the sun, that the difference of its attractions oil 



150 PRIMING AND LAGGING. 

the earth and the water is greater than the corresponding dif- 
ference in the case of the sun. It is computed that the effect 
of the moon in creating tides is about 2i times that of the sun. 

170. Combined Effects of both Sun and Moon. — Since each 
body, independently of the other, tends to raise the surface of the 
water at certain points, and to depress it at certain other points, 
the tides will evidently be higher when both bodies tend to raise 
the surface of the water at the same time, than when one tends 
to raise and the other to depress it. At new and full moon the 
two bodies act together, while at the first and the last quarter 
they act in opposition to each other. The tides at the former 
periods will therefore be the greater, and are called spring tides; 
and the tides at the latter periods are called neap tides. The ratio 
of the spring to the neap tide is that of (2J + 1) to (2i — 1), 
or of 5 to 2. 

The height of the tide is also affected by the change in the 
distance of the attracting body. For instance, when the moon 
is in perigee, the tides tend to run higher than when the moon is in 
apogee; and when the moon is in perigee, and also either new or 
full, unusually high tides will occur. 

171. Priming and Lagging of the Tides. — Each of these bodies 
may be supposed to raise a tidal wave of its own, and the actual 
high water at any place may be considered to be the result of 
the combination of the two waves. When the moon is in its 
first or its third quarter, the solar wave is to the west of the lunar 
one, and the actual high water will be to the west of the place 
at which it would have been had the moon acted alone. There 
is therefore at these times an acceleration of the time of high water, 
which is called the printing of the tides. In the second and the 
fourth quarter, the solar wave is to the east of the lunar one, 
and a retardation of the time of high water occurs, which is 
called the lagging of the tides. 

172. Although the theory of the tides, on the supposition that 
the earth is wholly covered with water, admits of easy explana- 
tion, the actual phenomena which they present are very much 
more complicated, and must be obtained principally from obser- 
vation. The lunar wave mentioned in the preceding article 
being grenter than the solar wave, we may consider the two to- 



ESTABLISHMENT. 151 

gether to constitute one great tidal wave, which at every moment 
tends to accompany the moon in its apparent diurnal path to- 
wards the west, raising the waters at successive meridians, but 
giving them little or no progressive motion. This tidal wave 
would naturally move westward with an angular velocity equal 
to that of the moon, so that at the equator its motion would be 
about 1000 miles an hour ; but the obstructions offered by the 
continents, the irregularity of their outlines, the uneven surface 
of the ocean bed, and the action of winds and currents and fric- 
tion, all combine not only to diminish the velocity of the tidal 
wave, but also to make it extremely variable. 

173. Establishment of a Port. — The interval of time between 
the moon's transit over any meridian and high water at that 
meridian varies at different places, and varies also on different 
days at the same place. This interval of time is called the 
lani-tidal interval. The mean of the values of this interval on 
the days of new and full moon is called the common establish- 
ment of a port. The mean of all the luni-tidal intervals in the 
course of the month is called the corrected establishment. These 
establishments are obtained by observation, and are given in 
Bowditch's Navigator, and also in other works. Thus the estab- 
lishment of Annapolis is 4h. 38m., and that of Boston llh. 12m. 
The time of transit of the moon over any meridian on any day 
can be obtained from the Nautical Almanac: and the sum of 
this time of transit and of the establishment of any port will be 
the approximate time of that flood tide which occurs next after 
the transit. Suppose, for instance, the time of high w T ater at 
Annapolis, on January 11, 1867, is desired. We have from the 
Almanac the time of the moon's transit, 4h. 33m. : adding to this 
the establishment, 4h. 38m. , the sum is 9h. 11m. This is, in this 
case, the time of the evening tide. The time of the morning tide 
may be obtained by subtracting 12h. 25m. from this time, which 
will give us 8h. 46m. a.m. A more accurate result in this 
last case might be obtained by taking from the Almanac the 
time of the preceding lower transit, and adding the establish- 
ment to it; but practically this would be a needless refinement, 
for the two results would only vary by about two minutes. 

The time of transit which the Almanac gives is in astronomical 



152 COTIDAL LINLS. 

time: hence the resulting time of high water will also be in astro- 
nomical time, and it will frequently happen that the time which 
we find, when turned into civil time, will fall on the civil day 
subsequent to the day for which the time is desired. Take, for 
instance, January 23, 1867, at Annapolis. The time of transit is 
January 23, 15h. 34m.: hence the time of high water is Janu- 
ary 23, 20h. 12m., or, in civil time, January 24, 8h. 12m. a.m. 
If, then, we wish the time of high water on the morning of the 
civil day January 23d, we must take from the Almanac the time 
of transit for the astronomical day of January 22d. 

There are other tables given in Bowditch's Navigator, and in 
the United States Coast Survey Reports, by which the time of 
high water can be obtained with greater accuracy than by the 
method given above. 

174. Cotidal Lines. — If the tidal wave were everywhere uni- 
form in its progress, it would come to all points on the same 
meridian at the same time. But, owing to irregularities induced 
by local causes, such is not the case, and places on different 
meridians often have high water at the same instant of time. 
Charts are therefore published on which are drawn lines con- 
necting places where high tides occur at the same instant: and 
these lines are called cotidal lines. These lines are usually ac- 
companied by numerals, which indicate the hours of Greenwich 
time at which high tides occur on the days of new and full moon 
along the different lines. 

175. Height of Tides. — At small islands in mid-ocean the height 
of the tides is not great, being sometimes less than one foot. When 
the tidal wave approaches a continent, and the w r ater begins to 
shoal, the velocity of the w T ave is diminished, and the height of 
the tide is increased. When the wave enters bays opening in 
the direction in which the wave is moving, the height of the tide 
is still further increased. 

The eastern coast of the United States may be considered to 
constitute three great bays : the first included between Cape Sable, 
in Nova Scotia, and Nantucket, the second between Nantucket 
and Cape Hatteras, and the third between Cape Hatteras and 
Cape Sable, in Florida. In each of these bays the tides, in gene- 
ral, increase in height from the entrance of the bay to its head. 



HEIGHT OF TIDES. 153 

For instance, in the most southern of these bays, the tides at 
Cape Sable and Cape Hatteras are not more than two feet in 
height; while at Port Royal, at the head of this bay, they are 
about seven feet. The same thing is noticed, in general, in 
smaller bays and sounds. For instance, in Long Island Sound, 
the height of the tide is two feet at the eastern extremity, and 
more than seven feet at the western. This increase of height is 
particularly noticeable in the Bay of Fundy, in which the height 
is eighteen feet at the entrance, and fifty and sometimes seventy 
feet at the head. 

There are in some cases, however, special causes which create 
exceptions to this general rule of increase of the tides between 
the entrance and the head of a bay. In Chesapeake Bay, for 
instance, which is wider at some places than it is at the entrance, 
and which lies about north and south, the tides in general dimin- 
ish in height as we ascend the bay. 

176. Tides in Rivers. — The same general principle holds good 
in the tides of rivers. "When the channel contracts or shoals 
rapidly, the height of the tide increases: when it widens or 
deepens, the height decreases. In a long river, then, the tides 
may alternately increase and decrease. For instance, at Tivoli, 
on the Hudson, between West Point and Albany, the tide is 
higher than it is at either of those two places. 

177. Different Directions of the Tidal Wave. — The tidal wave 
naturally tends to move towards the west ; but the obstructions 
offered by the continents and the promontories, and the irregular 
conformation of the bottom of the ocean, materially change the 
direction of its motion. Sometimes its direction is even towards 
the east. From a point about one thousand miles southwest of 
South America there appear to start two tidal waves, which travel 
in nearly opposite directions, one towards the west and the other 
towards the east. 

178. Four Daily Tides. — At some places the tides rise and 
fall four times in each day. This is ascribed to the existence of 
two different tidal waves, coming from opposite directions. This 
phenomenon occurs on the eastern coast of Scotland, where one 
wave comes into the North Sea through the English Channel, 
while a second wave comes in around the northern extremitv of 



154 TIDES IN LAKES. 

Scotland. At places where these two waves arrive at different 
times, each wave will produce two daily tides. 

179. Tides in Lakes and Inland Seas. — If there is any tide in 
lakes and inland seas, it is usually so slight as to be scarcely 
measurable. A series of careful observations has demonstrated 
the existence of a tide in Lake Michigan, which is at its height 
about half an hour after the moon's transit. The average height 
which it attains, however, is less than two inches. 

180. Other Phenomena. — A long the northern coast of the Gulf 
of Mexico there is only one tide in the day, the second one being 
probably obliterated by the interference of two waves. An ap- 
proximation to this state of things is noticed on the Pacific coast, 
where at times one of the daily tides has a height of several feet, 
and the other a height of only a few inches. A very curious 
statement is made by missionaries in reference to the tides at the 
Society Islands. They say that the tides there are uniform, not 
only in the height which they attain, but in the time of ebb and 
flow, high tide occurring invariably at noon and at midnight: so 
that the natives distinguish the hour of the day by terms de- 
scriptive of the condition of the tide. 



PLANTS, 155 



CHAPTER XII. 

THE PLANETS AND THE PLANETOIDS. THE NEBULAR 
HYPOTHESIS. 

181. The Planets and their Apparent Motions. — There are other 
celestial bodies besides the sun and the moon, which, while they 
share the common diurnal motion towards the west, appear to 
change their relative positions among the stars. These bodies are 
called planets, from a Greek word signifying wanderer. Some of 
them are visible to the naked eye, and some only become visible 
by the aid of a telescope. In some of them the change of position 
among the stars becomes apparent from the observations of a 
few nights : w T hile in others even the annual change of position 
is so small that they have for ages been considered to be fixed 
stars.* 

This change of position is determined, as it was determined in 
the case of the sun and the moon, by observations of their right 
ascensions and declinations. When such observations are made, 
the apparent motions of the planets are found to be very irregular. 
Sometimes they appear to move towards the east, and sometimes 
towards the w 7 est, w r hile at other times they appear to be station- 
ary in the heavens. Such irregularity in the direction of their 
motion is at once seen to be incompatible with the supposition 

* The times of meridian passage of all the planets which ever become 
conspicuously visible are given in the American Ephemeris. The altitude 
which any planet has when on the meridian is obtained from the corre- 
sponding zenith distance, and this is found at any place whose latitude 
is known, from the formula 

z — L — d : 
the declination being also given in the Ephemeris. It must be remembered 
that (L — d) becomes numerically (L-\-d) when L and d have different 
names ; that z in any case has the same name as (L — d) ; and that when z 
is north, the bearing of the body is south, and conversely. 



156 



PLANETS. 



that they are, like the moon, satellites of the earth, revolving 
about it as a centre. The next supposition that will naturally 
be made is that they may revolve about the sun. 

182. Heliocentric Parallax. — In order to test the correctness 
of this second supposition, we must first, from the apparent 
motions which are observed from the earth, deduce the corre- 
sponding motions which would be seen by an observer stationed 
at the sun. Fig. 64 will serve to show how, by means of a body's 
heliocentric parallax (Art. 56), the position which that body 
would have if seen from the sun's centre — in other words, its helio- 
centric position — may be determined from its geocentric position. 
In this figure let S represent the sun, ABCE the earth's orbit, 
the plane of which intersects the celestial sphere in the circle 




Fig. 64. 



VHD, and P the position of a planet when projected upon the 
plane of the ecliptic. Let the vernal equinox be supposed to lie 
in the direction EV or SV, which two lines must be supposed 
sensibly to meet when prolonged to the celestial sphere. Draw 
the lines EP and SP. Since the distances of the planet from the 
sun and the earth are finite, these lines will not lie in the same 
direction, and the angle EPS which they make with each other 
will be the difference of the directions in which the planet is seen 
from the earth and the sun : in other w r ords, the planet's helio- 



ORBITS OF THE PLANETS. 157 

centric parallax in longitude. Through S draw SK parallel to 
ED. The angle YEP, or its equal, VSK, is the planet's geo- 
centric longitude, and is obtained by observation. The sum of 
this angle and the angle EPS is the angle VSP, or the planet's 
heliocentric longitude. Provided, then, we know the angle EPS, 
we can readily obtain the angle VSP. Now, in the triangle PES 
we know from Kepler's Third Law the ratio of the sides SP 
and ES, which are the distances of the planet and the earth from 
the sun; and the angle PES, the planet's angular distance from 
the sun, or its elongation (Art. 56), can be obtained by observa- 
tion. The angle EPS may then be readily computed. 

By a similar method the planet's heliocentric latitude may be 
determined from its geocentric latitude, and the heliocentric 
place of a planet may thus be obtained at any time. 

183. Orbits of the Planets. — When the motions of the planets 
as they would be seen from the centre of the sun are thus deduced 
from their observed motions with reference to the earth, all the 
apparent irregularities of motion disappear. The planets are 
found to revolve from west to east in ellipses about the sun in 
one of the foci, the eccentricity of the ellipses diminishing, as a 
general rule, as the magnitude increases. The planes of the 
orbits are found to be nearly coincident with the plane of the 
ecliptic, and Kepler's and Newton's laws are exactly fulfilled in 
the case of each planet. The line in which the plane of each 
planet intersects the plane of the ecliptic is called the line of 
nodes, and the terms perihelion and aphelion have the same 
signification that they have in the case of the earth. 

184. Inferior and Superior Planets. — The planets are divided 
into two classes : inferior and superior planets. The inferior 
planets are those whose distances from the sun are less than the 
distance of the earth from the sun, and whose orbits are therefore 
included within the orbit of the earth. The inferior planets are 
Mercury and Venus.* The superior planets are those whose 

* There are some astronomers who are inclined to suspect the existence 
of a third inferior planet, Vulcan, distant about 13,000,000 miles from the 
sun. There is so much doubt about it, however, that it is not generally 
admitted into the list of planets. See North British Review, August, 1860. 



158 



INFERIOR PLANETS. 



distances from the sun are greater than that of the earth, and 
whose orbits therefore include the orbit of the earth. The 
superior planets are Mars, Jupiter, Saturn, Uranus, and Neptune. 
There is besides these a group of small planets, called minor 
planets, planetoids, or asteroids, situated between Mars and 
Jupiter. Up to September, 1871, 117 of these minor planets had 
been discovered. The earth is also a planet, lying between 
Venus and Mars. It is therefore a superior planet to Mercury 
and Venus, and an inferior planet to the other planets. Its 
sidereal period is greater than the periods of the inferior planets, 
and less than those of the superior planets. 

INFERIOR PLANETS. 

185. In Fig. 65 let 8 represent the sun, pp'p'P"" the orbit 
of an inferior planet, the plane of which is supposed to coincide 




Fig. 65. 



with the plane of the ecliptic, ABDE the orbit of the earth, and 
the circle KGH the intersection of the plane of the ecliptic with 
the celestial sphere. Suppose the earth to be at E. When the 
planet is at P, between the earth and the sun, or at P nf , on the 
opposite side of the sun to the earth, it has the same geocentric 



DIRECT AND RETROGRADE MOTION. 1-39 

longitude as the sun, and is in conjunction with it. The position 
at P is called the inferior conjunction, and that at P ,n the superior 
conjunction. 

The greatest angular distance of the planet from the sun will 
^evidently occur when the line connecting the planet and the earth 
is tangent to the orbit of the planet ; that is to say, when the planet 
is at P" or P"". The position at P" is called the greatest western 
elongation of the planet, that at P"" the greatest eastern elongation 
of the planet. We have already seen in Art. 92 that the rela- 
tive distances of the planet and the earth from the sun are at 
once obtained when we have measured the greatest elongation. 

186. Direct and Retrograde Motion. — The motion of the infe- 
rior planets is always in reality from west to east, or direct, as it 
is called; but when the planet is near its inferior conjunction, 
its motion is apparently from east to west, or retrograde. This 
apparent retrograde motion is explained in Fig. 65. Let the 
planet be at its inferior conjunction at P, and let both the earth 
and the planet move on about the sun in the direction EABD. 
The angular and the linear velocity of the planet about the sun 
being greater than they are in the case of the earth, when the 
earth arrives at E', the planet will be at some point P', and will 
lie in the direction E' G; the sun, on the other hand, will lie in 
the direction E'S' . While, then, the earth is advancing from E 
to E\ the sun and the planet appear to move away in opposite 
directions from the point C, on which both were projected when the 
earth was at E. But the apparent motion of the sun is invariably 
towards the east ; hence the planet has apparently moved towards 
the west. It may also be shown that the same apparent retro- 
grade motion occurs when the planet is approaching inferior 
conjunction, and is within a short distance of it. 

187. Stationary Points. — When the planet is at P"", it is 
moving directly towards the earth in the direction P""E, and 
the motion of the earth in its orbit gives the planet an apparent 
motion in advance. The same must also be the case when the 
planet is at P". Since then the motion of the planet is direct 
at the greatest elongations, and retrograde at inferior conjunction, 
there must be a point in the orbit between inferior conjunction 
and each elongation at which the planet neither advances nor 



160 ELEMENTS OF A PLANETS ORBIT, 

recedes, but appears stationary in the heavens. These points 
are called the stationary points. 

At all other parts of the orbit except those which have been 
discussed, the apparent motion of the planet is direct; but the 
velocity with which it moves is subject to great variation. It 
was on account of this irregularity, both in the direction and the 
amount of their apparent motions, that these bodies were called 
wandering stars by the ancient Greeks. 

188. Evening and Morning Stars. — Except at the times of 
conjunction, an inferior planet is either to the east or to the west 
of the sun. When it is to the east of the sun it will set after 
the sun has set, and when it is to the west of the sun it will rise 
before the sun has risen. In certain parts of its orbit the planet's 
elongation from the sun is sufficiently great to carry the planet 
beyond the limits of twilight, or 18° (Art. 101); it will then be 
an evening star if to the east of the sun, and a morning star 
if to the west. It is only to Venus, however, that these terms 
are commonly applied, at least so far as the inferior planets are 
concerned : since Mercury is so near to the sun that it is seldom 
visible, and even when it is visible, it appears like a star of only 
the third or the fourth magnitude. 

189. Elements of a Planet's Orbit — In order to compute the 
position in space at any time of either an inferior or a superior 
planet, we must be able to determine: — 

1st. The relative position of the plane of the planet's orbit to 
the plane of the ecliptic ; 

2d. The position of the orbit itself in the plane in which it 
lies; 

3d. The magnitude and the form of the orbit; and 

4th. The position of the planet in its orbit. 

These four conditions require the knowledge of seven distinct 
quantities. The first condition is satisfied if we know (1) the 
position of the line in which the plane of the orbit intersects the 
plane of the ecliptic, or, what amounts to the same thing, the 
longitude of the planet's nodes, and (2) the inclination of the 
two planes to each other. The second condition is satisfied if 
we know (3) the longitude of the perihelion. The third condi- 
tion is satisfied if we know (4) the semi-major axis, or the 



LONGITUDE OF THE SGDE. 



161 



planet's mean distance from the sun, and (5) the eccentricity of 
the orbit. Finally, the last condition is satisfied if we know 
(6) the time in which it makes one complete revolution about 
the sun, or its 'periodic time, and (7) the time when the planet 
is at some known place in its orbit, as, for instance, the peri- 
helion. These seven quantities are called the elements of the orbit. 
190. Heliocentric Longitude of the Node. — A planet is at its 
nodes when its latitude is zero; and if the heliocentric longitude 
of the planet at that time can be determined, it will also be the 
heliocentric longitude of the node, since the line of nodes of 
every planet passes through the sun. But the heliocentric longi- 
tude of a nlanet when at its node differs from the geocentric 
longitude (which may be obtained directly from observation), 
excepting only in case the earth itself happens at that time to 
be on the line of nodes. We must therefore be able to deduce 
the heliocentric longitude of the planet from the geocentric. 
When the planet's distance from the sun is known, this can be 
done by the method explained in Art. 182; and when this dis- 
tance is not known, the following method can be used. 

N 




Fig. 66. 

In Fig. 66, let S be the sun, PGHK the orbit of a planet, 
CDEE' the orbit of the earth, and NM the line of nodes of the 



162 INCLINATION OF A PLANET'S OiiBIT. 

planet. Let the vernal equinox lie in the direction EV or SV. 
Let E be the position of the earth when the planet is on the line 
of nodes at P. The elongation of the planet, or the angle PES, 
and the geocentric longitude of both planet and node, or the 
angle VEP, can be obtained by observation. Suppose both 
planet and earth to move on in their orbits, and the earth to be 
at E' when the planet again reaches the same node, and let the 
planet's elongation at this time, or the angle SE'P, be observed. 
Now, since the earth's orbit, although represented in the figure 
by a circle, is r 'ally an ellipse, ES and E'S will not in general 
be equal to each other. The value of each, however, can be 
readily obtained from the solar tables. The same tables will 
also give us the angle ESE\ which is the angular advance of the 
earth in its orbit in the interval. In the triangle ESE f , then, 
knowing two sides and the included angle, we can compute the 
side EE', and the angles SEE' and SE'E. The angles PES 
and PE'S having been obtained by observation, we can find the 
angles PEE' and PE'E. Then in the triangle PEE', knowing 
two angles and the included side, we can compute the side EP. 
Finally, in the triangle PES, knowing an angle and the two in- 
cluding sides, we can obtain PS, or the planet's distance from 
the sun, and the angle EPS, th? planet's heliocentric parallax, 
from wdiich, and the planet's observed geocentric longitude, we 
can obtain the planet's heliocentric longitude, as in Art. 182. 
This is, as we have already seen, the heliocentric longitude of 
the node. 

The nodes of every planet are found to have a westward 
movement, similar in character to the precession of the equi- 
noxes. It is a very slight movement, however, being only 70' a 
century in the case of Mercury, and being less than that in the 
case of the other planets. 

191. Inclination of the Planets Orbit to the Ecliptic. — The 
line of nodes of any planet being, as we have just now seen, a 
nearly stationary line in the plane of the ecliptic, the earth 
must pass it once in very nearly six months in its revolution 
about the sun. The inclination of the plane of any planet's 
orbit to the plane of the ecliptic may be determined by obser- 
vations made when the earth is on the line of nodes. In Fig. 




PERIODIC TIME. 163 

67 let E be the earth, and EN the line 
of nodes of a planet. Let AEN be the 
plane of the ecliptic, and P the position 
of a planet projected on the surface of 
the celestial sphere. With EP as a 
radius, let the arc PA be described, per- 
pendicular to the plane of the ecliptic, Fi s- 67 - 
and also the arcs PN and NA. In the spherical triangle PNA, 
right-angled at A, the arc PA, which measures the angle PEA, 
is the geocentric latitude of the planet; AN, or the angle AEN, 
is the difference between the geocentric longitudes of the planet 
and the node; and the angle PNA is the inclination of the plane 
of the orbit to the plane of the ecliptic. In the triangle PNA, 
we have," 

tan PA 
tan PNA = - — j-^- 
sin AJS 

It may be noticed in this formula that, since the sine of a small 
angle varies more rapidly than the sine of a large angle, an 
error in AN will affect the result the less, the greater AN itself 
happens to be at the time of observation : that is to say. the 
farther the planet is from the node. 

192. The Periodic Time of an Inferior Planet. — The time in 
which an inferior planet makes one complete revolution about 
the sun, or its periodic time, may be found by taking the interval 
of time between two successive passages of the planet through 
the same node. The accuracy of this method is however dimin- 
ished by the small inclination (less than 7°) of the planes of the 
orbits to the plane of the ecliptic, which renders it difficult to 
determine the instant when the latitude of the planet is zero. 
It is found to be better to determine the planet's syuodical period, 
or the interval between two successive conjunctions of the same 
kind, and from it to compute the sidereal period. The condi- 
tions of this problem, and the method by which it is solved, are 
identical with those in the case of the synodical and the sidereal 
period of the moon, Art. 141. The formula there given was, 

r ~ S + T 
In applying this to the case of an inferior planet, P and S 



164 MERCLUY. 

denote the sidereal and the synodical period of the planet, and 
T denotes, as before, the sidereal year. 

Instead of using the interval between two conjunctions as 
the synodical period, we may take the interval between two 
greatest elongations of the same kind. A very accurate mean 
synodical period is obtained by taking two elongations separated 
by a long interval, and dividing this interval by the number 
of synodical periods which it contains. 

193. It is hardly necessary to describe the methods by which 
the other elements given in Art. 189 are determined. It will 
readily be understood that the distance of the planet from the 
sun, obtained as in Art. 190, or by Kepler's Law, will enable 
us to obtain the form and the magnitude of the ellipse in which 
the planet moves. The method by which the longitude of the 
perihelion is obtained, although not intrinsically difficult, is too 
elaborate for this work. Finally, when the longitude of the 
perihelion is obtained, the time of the planet's perihelion passage 
may evidently at once be determined. The perihelion of Venus 
has a very minute retrograde motion : the perihelia of all the 
other planets have an eastward motion, similar to that of the 
earth's line of apsides. 

MERCURY. 

194. Mercury revolves about the sun at a mean distance of 
about 36,000,000 miles, the eccentricity of its orbit being about 
*th. Its synodical period is about 116 days, and its sidereal 
period 88 days. Its real diameter is about 3000 miles.* Its 
mass is a matter of considerable uncertainty, and quite different 
values of it are given by different astronomers. It may be 
assumed to be approximately equal to ygth of the mass of the 
earth. 

The greatest elongation of Mercury is only about 28° 20' : 

* The distances and the diameters of all the celestial bodies except the moon 
depend for their accuracy upon the accuracy with which the solar parallax 
is determined. An error of 1 7/ in this parallax would affect the sun's dis- 
tance from us to the amount of several millions of miles, and propor- 
tionally also the distances and the diameters of the other celestial bodies; 
the values given must therefore be considered to be approximate. 



Y r EXITS. 165 

and hence the planet is rarely visible to the naked eve, and U 
never a conspicuous object in the heavens. It is not generally 
believed that it has any atmosphere. It exhibits phases similar 
to those of the moon, and due to the same causes. It is asserted 
by some observers that it rotates on an axis in about 24 hours, 
though the truth of the assertion is by no means unchallenged. 
If it has any compression, it is extremely small. 

VENUS. 

195. The mean distance of Venus from the sun is about 
67,000,000 miles, its synodical period is 584 days, and its sidereal 
period 225 days. The eccentricity of its orbit is small, being 
only about T ^ B ths. Venus is nearly as large as the earth, its 
diameter being 7,600 miles. Its mass is about |ths of the mass 
of the earth. It has no perceptible compression. 

The greatest elongation of Venus from the sun amounts to 
about 47° 15', and hence it is often visible as an evening or a 
morning star. At certain times its brightness is so great that 
it can be seen in broad daylight with the naked eye, while at 
night shadows are cast by the objects which it illuminates. It 
exhibits phases similar to those of Mercury. 

It seems to be generally admitted that Venus has an atmos- 
phere the density of which is not very different from that of 
the earth's atmosphere. Observations of spots upon the disc 
go to show that the planet rotates upon an axis in a period of 
about 23] hours. 

It is suspected that Venus is attended by a satellite, but its 
existence is still a mooted point among astronomers. 

196. Transits of Venus. — We have already seen (Art. 93) 
how a transit of Venus across the sun's disc is employed in de- 
termining the distance of the earth from the sun. If the plane 
of the orbit of Venus coincided with the plane of the ecliptic, 
a transit would occur whenever the planet came into inferior 
conjunction, or once in every 584 days. Owing to the inclina- 
tion of the planes to each other, however, it is evident that at 
the time of inferior conjunction the planet may have too great a 
latitude to touch any part of the sun's disc. Now the phenomenon 
of a transit of Venus is analogous to a solar eclipse, and there- 



1G6 P TRANSITS OF VENUS. 

fore if in Fig. 60 we suppose M to* be Venus, the formula ob- 
tained in Art. 160 will apply equally well to the limits of the 
geocentric latitude of Venus within which a transit is possible. 
These limits are the sum of the semi-diameters of the sun and 
the planet and of the parallax of the planet, diminished by 
the parallax of the sun. The greatest value of the limits will 
be found to be about 17' 49". When, therefore, the latitude of 
Venus is more than 17' 49" at the time of inferior conjunction, 
no transit will occur : and as Venus in every sidereal revolution 
attains a latitude of over 3° 23', it is at once evident that a 
transit is only a rare occurrence. 

197. Intervals between Transits. — Since the latitude of Venus 
is so small when a transit occurs, it is plain that the planet must 
be either at or very near one of its nodes. Now, let us suppose 
that Venus is at its node at the time of inferior conjunction, 
under which circumstances a transit will, of course, take place. 
The sidereal period of Venus is 224.7 days. Now, we have, 

224.7d. X 13 = 2921.11d.; and, 

365.256d. X 8 = 2922.05d. 
At the end of eight years, then, Venus will be very near the 
same node at the time of inferior conjunction, and a transit will 
probably occur. Again, we have, 

224.7d. X 382 = 85835.4d. 
365.256d. X 235 = 85835.16d. ; 
so that transits at the same node also occur every 235 years. In 
the same way transits may also occur at the other node: and the 
intervals between transits at either node are found to be 8, 105 i, 
8, 121 J, 8, &c. years.* The longitude of the ascending node of 
Venus is 75° 20', and a transit at that node must occur, when it 
occurs at all, at the time when the earth is near that point, which 
is about the 6th of June. For a similar reason transits at the 
descending node occur about the 6th of December. 

The last transit occurred in June, 1769 ; the next two will 
occur in December, 1874, and December, 1882. After that no 
more will occur until June, 2004. 

* Thus the years of transits at the ascending node are 1761, 1769, and 
2004; at the descending node, 1639, 1874, and 1882. 



SUPERIOR PLANETS. 



167 



SUPERIOR PLANETS. 

198. The superior planets are, as they have already been de- 
fined to be, planets whose orbits include the orbit of the earth. 
They have, like the inferior planets, superior conjunction, but 
can evidently have no inferior conjunction. Their elongation 
from the sun, eastern and western, can have all values between 0° 
and 180°. When their elongation is 90°, they are said to be in 
quadrature, and when it is 180°, in opposition. Much that has al- 
ready been said in this chapter in reference to the inferior planets 
is equally true in reference to the superior planets. The elements 
of the orbits are in both instances the same, and so are the me- 
thods by which the heliocentric longitudes of the nodes and the 
inclinations of the orbits are determined. In some other points 
there is a difference between the two classes of planets ; and these 
points w r e shall now proceed to examine. 

199. Retrograde Motion. — The superior planets, like the infe- 
rior planets, have at times an apparent retrograde motion, which 

jsr 




M'MM"M" 

Fig. 68. 

occurs at or near the time of opposition. The explanation of 
this retrogradation will be seen by a reference to Fig. 68. S is 



168 PERIODS OF A SUPERIOR PLANET. 

the sun, EE'E"E m the orbit of the earth, and CGDP the orbit 
of a superior planet, the plane of which is supposed to coincide 
with the plane of the ecliptic, and to meet the celestial sphere in 
the circle ANBM. Let the earth be at E, and the planet at P, 
180° in geocentric longitude from the sun. The planet will ap- 
pear to be projected upon the celestial sphere at the point M. 
Let both earth and planet revolve in their orbits in the direction 
indicated by the arrow. When the earth has reached the point 
E f , the planet, whose angular velocity is less than that of the earth, 
will have reached some point P', and will lie in the direction E'M f : 
in other words, it has apparently retrograded. If, on the other 
hand, the earth is at E" and the planet at P, or in superior con- 
junction, the apparent motion of the planet is at that point 
direct, and its angular velocity appears to be greater than it 
really is; for when the earth is at E'", and the planet at P', the 
latter, having in reality moved through the arc MM" since con- 
junction, appears to have moved through the arc MM'". 

The apparent motion, then, of a superior planet is direct in 
all cases except when it is at or near its opposition. The ap- 
parent motion of an inferior planet has been shown to be retro- 
grade at and near the time of inferior conjunction (Art. 186). 
Now, since the earth is a superior planet to an inferior planet, 
and an inferior planet to a superior planet, we see, by comparing 
the two cases, that the retrograde motion occurs in each class 
of planets at and near the time when the inferior planet comes 
between the sun and the superior planet. 

The stationary points in the orbit of a superior planet are 
identical in character w T ith those in the orbit of an inferior 
planet, and occur when the retrograde motion is changing to the 
direct motion, or the direct to the retrograde. 

200. Synodical and Sidereal Periods. — The synodical period 
of a superior planet is the interval of time between two suc- 
cessive conjunctions or tw T o successive oppositions. When a planet 
is in conjunction with the sun, it is above the horizon only in 
the day time ; but when it is in opposition, it is above the horizon 
during the night, and can therefore be readily observed. Hence 
in obtaining the synodical period it is better to employ the in- 
terval of time between two oppositions : and by determining the 



DISTANCE FROM THE SUN. 169 

times of two oppositions which are not consecutive, and dividing 
the interval between them by the number of synod ical revo- 
lutions which it contains, we may obtain a mean value of the 
synodical period. By using times of opposition which were ob- 
served and recorded before the Christian era, a very accurate 
value of the mean synodical period may be obtained. 

The method of deducing the periodic time from the synodical 
period is the same that was used in the case of the inferior 
planets (Art. 192), with the important exception that in the 
present case it is the earth that gains 360° upon the planet in the 
course of a synodical revolution, and not the planet that gains it 
upon the earth. If, therefore, we denote the periodic times of 
the earth and the planet by Tand P, and the synodical period of 
the planet by S, we shall have (Art. 141), 
36(T _ 36(T __ 36(T 
T P ~ 8 

■ P = ST ■ 
' S — T ' 

which gives the value of any planet's sidereal period in terms of 
its synodical period and the sidereal year. 

201. Distance of a Superior Planet from the Sun. — The distance 
of a superior planet from the sun may be obtained by the method 
of Art. 190: or it may be obtained from observations made at 
the time it is in opposition. In 
Fig. 69, let S be the sun, E the 
earth, and P a superior planet 
at the time of opposition, the 
planes of the two orbits being 
supposed to coincide. At the 
end of a short interval, let the earth have moved to E f , and the 
planet to P'; the angle E' OE will be the amount of the apparent 
retrogradation of the planet in that time. The periods of the 
earth and the planet being known, we can compute the angular 
advance of each planet in the given interval, thus obtaining the 
angles E'SE and P r SP. The radius vector of the earth's orbit, 
SE', can be found from the Nautical Almanac. Then, in the 
triangle E'SP', we know the side E'S, the angle E'SP', which is 
the angular gain of the earth on the planet, and the angle E'P'S, 




170 MARS. 

which is the sum of P'SO, or the advance of the planet, and 
P f OS, or the apparent retrogradation of the planet. We can 
therefore compute the side SP', which is the distance required. 

If we suppose the real distance of the sun from the earth not 
to be known, this method will still give us the ratio between 
this distance and the distance of the planet from the sun. And 
furthermore, if we can determine the real distance of the planet 
from the earth by observations of its parallax at the time of 
opposition (when it is nearest to the earth), by obtaining its 
displacement in right ascension when far east and far west of the 
same meridian, we can readily obtain the real distance of the 
earth from the sun. This has recently been done in the case of 
the planet Mars, and the distance of the earth from the sun 
has been deduced, as already given (Art. 94). 

202. Evening and Morning Stars. — The angular velocity of a 
superior planet towards the east is less than that of the earth, 
and consequently also less than the sun's apparent angular 
velocity in the same direction. After conjunction, therefore, the 
planet will lie to the west of the sun, and its elongation will 
continually increase. When this elongation exceeds about 30°, 
the planet will begin to be visible as a morning star, and will 
so continue until it has fallen 180° to the west of the sun, and 
is in opposition. It will then rise about sunset and set about 
sunrise. After the time of opposition it will lie more than 180° 
to the west of the sun, or, what is the same thing, less than 180° 
to the east of it, and will rise before sunset. It will therefore 
be an evening star from opposition to conjunction. 

MARS. 

203. The synodical period of Mars is 780 days, and its side- 
real period 687 days. Its mean distance from the sun is 
141,000,000 miles, and the eccentricity of its orbit about y T th. 
Its diameter is 4,900 miles. Different values are given to its 
compression, varying between the limits of -g-^th and ygth. As 
it is attended by no visible satellite, its mass is a matter of some 
uncertainty. It is probably about ith of that of the earth. 

204. Phases. — At opposition and conjunction the same hemi- 
sphere is turned towards both the sun and the earth, and con- 



MINOK PLANETS. 171 

sequently the planet appears full. At quadrature it appear? 
slightly gibbous. It is the only one of the superior planets 
which exhibits any sensible phases, excepting possibly Jupiter. 

Mars shines with a red light, and at orjposition is a very con- 
spicuous object, sometimes equalling Jupiter in brilliancy. 

205. Rotation, &c. — When examined in a telescope, the sur- 
face of Mars is seen to be covered with patches of a dull red- 
dish color, which are supposed to be land, interspersed with 
spots of a bluish or greenish hue, which are supposed to be 
water. By observation of these spots Mars is found to rotate 
upon an axis once in about 24J hours. The axis of rotation 
is inclined at an angle of 61° 18' to the plane of the ecliptic, 
and hence there must be a change of seasons not very different 
from the change which takes place on the earth. White spots 
are seen near the poles, w 7 hich decrease in the Martial summer 
and increase in the winter. These spots are supposed to be snow. 
The existence of an atmosphere of a moderate density is gene- 
rally admitted by astronomers. 

THE MINOR PLANETS. 

20G. Bode's Law. — In 1778 the astronomer Bode, of Berlin, 
announced (though he did not discover) the following curious 
relation between the distances of the different planets from the 
sun. The statement of this relation usually goes by the name 
of "Bode's law." If we take the series of numbers 

0, 3, 6, 12, 24, 48, 96, 192, 384, 
each of which, except the second, is double the preceding one, 
and add 4 to each of these numbers, the resulting series, 

4, 7, 10, 16, 28, 52, 100, 196, 388, 
will approximately represent the relative distances of the planets 
from the sun.* Thus Mercury is 36,000,000 miles from the 
sun, and Venus 67,000,000 miles; and these distances are to each 
other nearly in the ratio of 4 to 7. There was however (the 

* The Inst two numbers were not in the series as originally announced 
by Bode, since Uranus and Neptune had not then been discovered. The 
real distance of Neptune is one-fourth less than it should be, if this law 
were anything more than a coincidence. 



172 MINOR PLANETS. 

minor planets being then undiscovered) a break in the series, 
there being no planet corresponding to the number 28; and 
Bode ventured to predict that another planet might be found to 
exist at that point of the series : that is to say, between Mars and 
Jupiter. A similar prediction was made by Kepler, about the 
beginning of the seventeenth century. 

207. Discovery of the Minor Planets. — In 1800, six European 
astronomers formed an association for the express purpose of 
searching the heavens for new planets; and within the next six 
years four minor planets were discovered. These were named 
Ceres, Pallas, Juno, and Vesta. No more were discovered until 
the end of 1845, but since that time some have been discovered 
in nearly every year. The number discovered up to September, 
1871 (including Ceres, &c), was 117. 

The mean distances of these bodies from the sun vary from 
200,000,000 to 300,000,000 miles. They are all very small, 
the largest being probably not over 300 miles in diameter, and 
many of the others being too small to admit of measurement. 
Vesta is the only one which is ever visible to the naked eye, and 
its visibility is very rare. Some of the others are so small that 
they can scarcely be seen with the strongest telescope, even at 
opposition. 

The French astronomer Le Verrier has concluded that the 
mass of these 117 planetoids is by no means sufficient to produce 
the perturbations in the orbit of Mars and in that of Jupiter 
which are believed to be due to the attractions of this group. 
It is therefore extremely probable that many other planets, 
hitherto undiscovered, belong to the same cluster. 

208. Olbers's Theory. — Shortly after the discovery of the first 
four minor planets, Dr. Olbers advanced the theory that these 
planets were fragments of a single planet, which had been broken 
in pieces by volcanic action or by some other internal force. This 
theory is still in favor with many astronomers; others, however, 
object to it on the ground that if these bodies did formerly con- 
stitute one single body, their orbits ought to have a common 
point of intersection, which is very far from being the case. There 
are, however, certain striking resemblances in their orbits, which 
seem to argue something common in their origin. 



JUPITER. Li 6 

JUPITER. 

209. Jupiter is the largest planet of our system. At times it 
surpasses Venus in brilliancy, and even casts a shadow. It re- 
volves about the sun at a mean distance of 481,000,000 miles. 
The eccentricity of its orbit is about ^ih. Its synodical period is 
399 days, and its sidereal period 4333 days, or about 11.9 years. 
Its diameter is about 89,000 miles, and its volume is about 1400 
times that of the earth. It rotates on an axis in a little less 
than ten hours, and has a compression of y^th. Its phases are 
so slight as to be scarcely perceptible. 

210. Belts. — When examined through a telescope, the disc of 
Jupiter is seen to be streaked with dark belts, lying nearly 
parallel to the plane of its equator. With powerful telescopes 
these belts are found to have a gray or brown tinge. They are 
sometimes nearly permanent for several months, and sometimes 
they change their shape materially in the course of a few minutes. 
There are usually one broad and several narrower belts on each 
side of Jupiter's equator. 

It is generally supposed that Jupiter is surrounded by a dense 
atmosphere, and that these belts are fissures in this atmosphere, 
through which the dark body of the planet is seen. The distri- 
bution of the atmosphere in lines so nearly parallel to the equator 
is supposed to be due to currents in the atmosphere, similar in 
character to our trade- winds, but having a more decided easterly 
and westerly tendency, from the more rapid rotation and the 
greater size of the planet. A point on Jupiter's equator rotates 
with a velocity of about 28,000 miles an hour, while a point on 
our own equator rotates with a velocity of only about 24,000 
miles a day. 

211. Satellites. — Jupiter is attended by four satellites or moons, 
revolving about it from west to east. They are distinguished 
from each other by the numbers 1, 2, 3, and 4, the first satellite 
being the nearest to Jupiter. The second satellite is about as 
large as our moon, and the others are somewhat larger. They 
are not usually visible to the naked eye. though a few instances 
to the contrary are on record. The distance of the first satellite 
from Jupiter is 270,000 miles, and that of the fourth is 1,200,000 



174 SATELLITES OF JUPITER. 

miles. The first revolves about Jupiter in a period of 42 hours, 
and the fourth iu a period of 16d. 17h. 

212. Phenomena Presented by the Satellites. — The satellites, in 
the course of their revolution about their primary, present four 
distinct classes of phenomena, which are shown in Fig. 70. In 
this figure let S be the disc of the sun, and EE f E"E'" the orbit 




Fig. TO. 



of the earth. Let ./ be Jupiter, and ABDG the orbit of one of 
its satellites. Since the planes of all the orbits very nearly 
coincide with the plane of the ecliptic, we may consider ABDG 
to lie in that plane. Suppose the earth to be at E, and, in order 
to simplify the case, suppose it also to remain at that point 
during the short time required by the satellite to revolve about 
Jupiter. 

An eclipse of the satellite will occur when it passes through 
the arc MN, since it is then within the shadow formed by lines 
drawn tangent to the disc of the sun and that of Jupiter. It may 
readily be calculated that the length of the shadow, from J to C\ 
is about 55,000,000 miles, so that the shadow extends far beyond 
the orbit of the fourth satellite. In extremely rare cases this 
satellite, owing to the inclination of its orbit to the ecliptic, may 
fail to be eclipsed. 

An occiiltation of the satellite will occur when it passes through 
the arc AB, since it is then within the cone formed by lines 
drawn from E tangent to the disc of Jupiter. 

A transit of the shadow will occur when the satellite passes 
through the arc OH, its shadow being then cast upon the disc 
of Jupiter, and moving across it as a small round spot. 



VELOCITY OF LIGHT. 175 

Finally, a transit of the satellite will occur when it passes 
through the arc KL. 

It will evidently depend on the relative situations of the sun, 
the earth, and Jupiter, whether all these phenomena will be 
observed or not. When the earth, for instance, is at E f or E" ', 
it is plain that only an occultation and a transit will occur. 

The relative situations of the satellites to each other and to 
their primary are constantly changing. Sometimes all are on 
the same side of the primary : sometimes only one is visible, 
and sometimes, though very rarely, all are invisible. The times 
at which these different phenomena will occur are computed 
beforehand, and are given in the American Ephemeris, the time 
used being that of the meridian of Washington. The longitude 
of any place can therefore be obtained, at least approximately, 
by observations of these phenomena. 

213. Velocity of Light. — If the transmission of light is not in- 
stantaneous, it is evident that the same phenomenon, if observed 
both at E' and E 1 " (Fig. 70), will not occur at the same absolute 
instant of time at both places, but will occur later at E ,n by 
the time required for light to cross the orbit of the earth, a dis- 
tance of 185,000,000 miles. And such is actually the case. 
This peculiarity was first noticed by Horner, a Danish astrono- 
mer, in 1675, who found that the times at which the phenomena 
occurred were earlier by about eight minutes at E\ and later by 
the same amount at E nr , than the times computed for the mean 
distance of Jupiter from the sun. The time required by light in 
passing from E' to E nf has been found by observation to be very 
nearly 16m. 27s.: whence the velocity of light is calculated to be 
187,000 miles a second, a result agreeing very closely with the 
velocity obtained from the constant of aberration, discussed in 
Chapter VIII. 

214. Mass of Jupiter. — The mass of Jupiter is much more 
accurately known than the mass of any of the planets which 
have hitherto been described. The reason is that Jupiter is 
attended by satellites whose distances from their primary, and 
whose periods of revolution, can be obtained by observation. 
We are thus enabled to compare directly the attraction which 
Jupiter exerts on one of its satellites with the attraction which 



176 SATURN. 

the sun exerts on Jupiter; and as, by the law of gravitation, the 
ratio of these two attractions is directly as the ratio of the masses 
of the two attracting bodies, and inversely as the square of 
the ratio of the distances through which these attractions are 
exerted, it is evidently within our power to obtain the ratio of the 
two masses. Since the attraction of the sun on Jupiter is equal to 
the centrifugal force of Jupiter in its orbit, if we denote the dis- 
tance of Jupiter from the sun by D and its sidereal period by T, 
we have, by the formula of Art. 69, the expression for the sun's 

4 tz 2 D 

attraction on Jupiter equal to . In the same way, denoting 

the distance of a satellite from Jupiter by d, and its period by t, 

4 ~ 2 d 
we have for the attraction of Jupiter on the satellite, — — : so 

z 

Dt 2 
that the ratio of the two attractions is — . Finally, denoting 

the mass of the sun by Jlf, and that of Jupiter by m, we shall 
have> 

dT 2 ~ m X ~B' ; 
M __ DH 2 
•'* m — d 3 T 2 " 

By this formula an approximate value can be obtained of the 
mass of any planet which is attended by a satellite. 

The mass of Jupiter is found to be y^^th of that of the sun, 
or about 312 times that of the earth. 

SATURN. 

215. Saturn is, next to Jupiter, the largest planet of our sys- 
tem, and may fairly be considered to be the most interesting. 
It revolves about the sun at a mean distance of 881,000,000 
miles, in an orbit w T hose eccentricity is about jyth. Its synodical 
period is 378 days and its sidereal period 29.45 years. Its 
diameter is 72,000 miles, and it has a compression of about yoth. 
It is attended by eight satellites, the planes of whose orbits, with 
one exception, very nearly coincide with the plane of its equator. 
Two of these satellites are rarely visible in any but the strongest 



RINGS OF SATURN. 177 

telescopes. Their distances from the planet range from 121,000 
miles to 2,300,000 miles, and their sidereal periods from 22 hour3 
to over 79 days. With one exception, they appear to be smaller 
than our moor, 

216. Rotation, &c. — Saturn rotates upon an axis which is in- 
clined at an angle of about 62° to the plane of the ecliptic, in a 
period of 10 \ hours. Belts are seen on the body of the planet, 
similar to those of Jupiter, although less marked. Other indi- 
cations of the existence of an atmosphere have also been observed. 
The mass of the planet, determined by the motions of its satellites, 
is about 3 5Vo*h of that of the sun, or about 93 times that of the 
earth. 

217. Rings of Saturn. — When observed through a telescope, 
Saturn is seen to be surrounded by a marvellous system of lumi- 
nous rings, lying one within another in the plane of the planet's 
equator, and very nearly concentric with the planet. Although 
the planet itself was known to the ancients, the existence of these 
rings was not suspected until the seventeenth century. They were 
then supposed to be two rings, one within the other ; but later obser- 
vations, with improved instruments, have made it almost certain 
that the exterior of these two rings is itself composed of two, and 
it is probable that similar subdivisions also exist in the interior 
ring. A third independent ring, lying within the others, was 
discovered in 1850, by Professor Bond, at the Observatory of 
Harvard College. 

Considering only the two rings which were first discovered, 
and neglecting the subdivisions that may exist in each, the ap- 
proximate dimensions of the rings are as follows : — 

Outer diameter of exterior ring 170,000 miles. 

Breadth of exterior ring 10,000 miles. 

Distance between the two rings 2,000 miles. 

Outer diameter of interior ring 1-16,000 miles. 

Breadth of interior ring 17,000 miles. 

Distance of interior ring from Saturn 20,000 miles. 

218. The thickness of the rings is very small. Sir John 
Herschel estimated it at not more than 250 miles, while Professor 
Bond considered it to be only about 40 miles. The rings appear 
to rotate about the planet from west to east, the period of rotation 

12 



178 RINGS OF SATURN. 

being about 10J hours, according to Herschel. Various theories 
have been advanced as to the composition of these rings. One 
theory is that they are a collection of meteoric bodies, revolv- 
ing about the planet precisely as similar rings or groups of 
meteoric bodies are found to revolve about the sun. (Chap. XIII.) 
A second theory is that they are composed of nebulous matter, 
and still a third is that they are solid. According to Professor 
Bond, however, they are in a fluid state; and there are many 
considerations which favor this view. 

The main cause of the stability of the rings in reference to 
the planet is undoubtedly the centrifugal force induced by the 
rapid rotation above mentioned. Professor Peirce, in developing 
the theory proposed by Professor Bond, maintains that the per- 
manence of the equilibrium is finally dependent upon the attrac- 
tions exerted by the satellites upon the rings. Herschel says 
that unless they were originally adjusted in their present position 
with the minutest precision, they must have been gradually 
formed under the influence of all the existing forces. 

219. The rings must present a magnificent spectacle to the 
inhabitants of the planet. To an observer on Saturn's equator 
they will appear as an arch, passing through the zenith, and 
through the east and the west point of the horizon. To such an 
observer only the edge of the rings is visible. As he moves 
away from the equator, the altitude of the rings decreases, and 
the side of the rings becomes visible, presenting an appearance 
not unlike the familiar one of the rainbow. Under the most 
favorable circumstances of position, the rings will be projected 
against the sky as an arch with the enormous apparent diameter 
of about 15°, which is about 30 times the diameter which the 
sun presents to us. 

220. Disappearance of the Rings. — As Saturn revolves about 
the sun, the plane of its rings remains, like the plane of the 
earth's equator, fixed in space, and intersects the plane of the 
ecliptic in a line which is called the line of nodes of the rings. 
In Eig. 71, let S be the sun, ABCD the orbit of the earth, and 
EHLN the orbit of Saturn. Let HN be the line of nodes of 
the ringo, and draw the lines GO and KM parallel to HN t and 
tangent to the earth's orbit. When the planet is at H, the plane 



RINGS OF SATURN. 



179 



of the rings passes through the sun, and only the edge of the 
rings is illuminated. In such a case the rings will disappear, or 
at all events will only be seen, in very powerful telescopes, as 
an exceedingly narrow line. Furthermore, if, while the planet 
is within the lines GO and KM, the earth encounters the plane 




of the rings, they will again disappear. And thirdly, if, while 
the planet is within the same limits, the plane of the rings 
passes between the earth and the sun, the dark side of the rings 
will be turned towards the earth, and they will disappear, 
When the planet is beyond these limits, it is evident that the 
rings will always be visible, and will present an elliptical appear- 
ance, as represented at JE 

Now, we can readily compute the length of time which Saturn 
requires in passing through the arc GK. For in the triangle 
CSK, right-angled at C, we know the sides OS and KS, or the 
distance of the earth from the sun and that of the planet, and can 
therefore obtain the angle CKS. It will be found to be about 6° 1'. 
This angle is equal to the angle KSH, and therefore double 
this angle, or 12° 2', is the angle through which Saturn moves 
about the sun in passing through the arc GK. Now we know 
that Saturn makes a complete revolution about the sun in 



180 URANUS. 

10,759 days; and therefore we may find by a simple proportion 
the time which it requires to pass through 12 ° 2'. This time 
is found to be 359.6 days, or very nearly a sidereal year; so that 
the earth makes very nearly one complete revolution about the 
sun while Saturn is passing through the arc GK. 

221. Number of Disappearances. — Since Saturn's period of 
revolution is 29.45 years, these disappearances will occur at in- 
tervals of a little less than 15 years. Since the time during 
which the planet remains within the limits GO and KM is only 
six days less than a year, and since the earth may encounter the 
plane of the ring at any point in its orbit, it is almost certain 
that one such meeting will occur, and under certain circum- 
stances there may be three. Suppose, for instance, that the 
earth is at a when Saturn is at G, and that both bodies move 
about the sun in the direction EHLN. The earth will meet 
the plane of the rings somewhere in the arc a A, and the 
rings will disappear. The rings will continue to be invisible 
for some time, since their plane will lie between the earth 
and the sun. The earth will overtake the plane before Saturn 
reaches the point H, and after that time the rings will be visible 
until the planet is at H, when the plane passes through the sun, 
and the rings again disappear. The earth will now be near the 
point b, and the rings will continue to be invisible, since their 
dark side is turned to the earth. The earth, passing through C, 
will again meet the plane somewhere in the arc CD, and after 
that time the rings will be visible. No more disappearances 
will occur for about fifteen years, at the end of which interval 
the planet will pass through the arc 310. 

The last disappearance took place in 1862; the next will take 
place in 1877. 

URANUS. 

222. All the planets which have thus far been described, 
except of course the minor planets, were known to the an- 
cients; but the last two are among the comparatively recent 
discoveries of astronomers. Uranus was discovered by Sir 
William Herschel, in 1781, by pure accident. Herschel, as 
well as other astronomers whose attention was directed to it, 



URANUS. 131 

at first supposed it to be a comet; and it was only after 
several months of observation that it was found to be a planet. 
Several names were suggested for it, but the name of Uranus 
was finally adopted. The astronomical symbol for it which 
the English have adopted is formed from the initial letter of 
Herschel's name. 

Upon searching the star catalogues and other astronomical 
records, it was found that the planet had been observed no less 
than twenty times in the preceding 90 years, and had been con- 
sidered to be a fixed star, its daily motion being so slight as to 
have escaped notice. Indeed its period is so great that even its 
annual change of position is only a few degrees. 

223. These previous records, however, were of great assistance 
to astronomers in the determination of the elements of the 
planet's orbit. The sy nodical period of the planet is 370 days, 
and the sidereal period 30,687 days, or nearly 84 years. Its dis- 
tance from the sun is 1,800,000,000 miles, and its diameter 33,000 
miles. It is barely visible to the naked eye at opposition. 

It is attended by eight satellites, which are only visible in the 
most powerful telescopes ; and by means of their movements the 
mass of the planet is found to be about 13 times that of the 
earth. One very remarkable point about these satellites is that 
their motion about their primary is retrograde, or from east to west, 
in planes inclined about 79° to the plane of the ecliptic, while 
the motions of all the other satellites which have hitherto been 
described are from west to east, and in planes making very small 
angles with the plane of the ecliptic. It is proper to notice here 
that some astronomers doubt the existence, or rather the dis- 
covery, of more than four of these satellites. 

Scarcely more can be said of the physical appearance of 
Uranus than that it is uniformly bright. It exhibits neither 
spots nor belts, and therefore nothing can be determined as to 
any axial rotation, which is a question of special interest in the 
case of this planet, since one of the supports on which what is 
called "the nebular hypothesis" rests (Art. 228; is the assump- 
tion that the satellites of every planet revolve about it in the 
same direction in which it rotates on its axis. 



182 NEPTUNE. 



NEPTUNE. 

224. Early in the present century the conviction forced itself 
upon the minds of many astronomers that there must exist still 
another planet, exterior to Uranus. The circumstance which 
led to this conclusion was the existence of irregularities in the 
orbit of Uranus, over and above the irregularities which were 
due to the attractions exerted by the planets then known. The 
first systematic attempt to deduce the elements of the orbit of 
this unknown planet from these irregularities seems to have been 
made by Mr. Adams, of England, in 1843-5. The position which 
he assigned to the planet was in heliocentric longitude 329° 19', 
but this determination was not then made public. The same 
intricate problem was also solved by M. Le Verrier, of Paris, in 
1845-6, and the longitude which he obtained was 326°. During 
the summer of 1846 search was made for the planet in England, 
but without success, owing to the want of a proper star-map. 
The observatory at Berlin, however, was better supplied ; and on 
the night of September 23d, in compliance with a request made 
in a letter received that day from Le Verrier, Dr. Galle at once 
detected, in longitude 326° 52', what was apparently a star of 
the eighth magnitude, though it was not laid down on the map. 
Subsequent observations showed that this body was really a 
planet, and it was agreed to give it the name of Neptune. 

"Such," in the words of Hind, "is a brief history of this most 
brilliant discovery, the grandest of which astronomy can boast, 
and an astonishing proof of the power of the human intellect." 

225. The synodical period of Neptune is 367 days, and its 
sidereal period 60,127 days, or about 164 years. Its mean dis- 
tance from the sun is 2,800,000,000 miles, and its diameter 37,000 
miles. It is attended by one satellite, and some astronomers 
suspect the existence of a second. The mass of Neptune is about 
seventeen times that of the earth. The planet is not visible to 
the naked eye. 

Nothing is yet determined as to the physical appearance or 
the axial rotation of the planet. A remarkable circumstance in 
connection with the satellite is that, like the satellites of Uranus, 
it moves about its primary from east to west 



NEBULAE HYPOTHESIS. 18? 

226. It may help us in our conception of the immense distance 
of Neptune from us, even when it is in opposition, to consider 
that light, with its velocity of 186,000 miles a second, requires 
four hours to come from the planet to the earth. If there are 
any inhabitants of Neptune, the sun will to them have an appa- 
rent diameter of only ^th of what it has to us, since the distance 
of Neptune from the sun is about thirty times that of the earth. 
It will therefore appear to them only about as large as Venus 
appears to us, under the most favorable circumstances. Saturn, 
Jupiter, and Uranus may possibly be visible to them as extremely 
small bodies, but it is very doubtful if any of the other planets 
of our system are visible, even with the strongest telescopes. 
We shall see hereafter, in the Chapter on Fixed Stars, that, with 
a base-line of even 185,000,000 miles (the diameter of the earth's 
orbit), attempts to determine the distances of the stars are, as a 
general rule, wholly unsuccessful ; but it is very likely that similar 
attempts made by observers on Neptune, with a base-line thirty 
times as long as ours, may give more satisfactory results. 

227. Relative Sizes and Distances of the Planets. — The relative 
distances of the planets from the sun, their relative magnitudes, 
as well as other numerical data concerning them, will be found 
in tables in the Appendix. In Plate I. will be seen a represen- 
tation of their relative magnitudes, as they would appear to an 
observer stationed at the same distance from all of them. 

THE NEBULAR HYPOTHESIS. 

228. Points of Resemblance in the Planetary Phenomena. — The 
light of the planets and the satellites, when examined in the 
spectroscope, produces only the ordinary spectrum of reflected 
solar light. While, therefore, the spectral analysis of the light 
of the sun, the stars, and other heavenly bodies which shine by 
their own light, enables us to determine to some extent the ele- 
ments of which they are composed, a similar experiment tells 
us nothing of the constitution of the planets or the satellites. 
What we do know, however, of their form, their appearance, 
their mass, and their density, leads us to conclude that they are 
bodies not dissimilar to the earth in general constitution. There 
are, besides, certain remarkable coincidences in the various phe- 



184 NEBULAR HYPOTHESIS. 

nomena exhibited by the sun, the planets, and the satellites, 
which seem to point to a common origin of the whole solar sys- 
tem. The principal of these coincidences are the following: — 

(1.) All the planets revolve about the sun in the same direction 
in w r hich the sun rotates upon its axis : that is to say, from west 
to east. 

(2.) The planes of the planetary orbits nearly coincide with 
the plane of the sun's equator. 

(3.) The satellites of each planet, as far as known, revolve 
about their primary in the same direction in which the primary 
rotates upon its axis. The satellites of Uranus and Neptune 
may or may not form an exception to this rule, for these planets 
are so distant that observation fails to detect any axial rotation 
in them. 

(4.) The planes of the orbits of the satellites of each planet 
approximately coincide with the plane of that planet's equator. 

(5.) Both planets and satellites revolve in ellipses of small 
eccentricity. 

229. The Nebular Hypothesis. — The idea of the nebular hypo- 
thesis seems to have presented itself at about the same time to 
both Sir William Herschel and Laplace. The principal points 
in it are the following. All the matter w T hich now composes the 
sun, the planets, and the satellites once existed as a single nebu- 
lous mass, extending beyond the present orbit of Neptune, and 
rotating on an axis from west to east. In the progress of ages 
this nebulous mass slow T ly contracted and condensed, from the 
loss of the heat which it radiated into space, and from the 
gravitation of its particles towards the centre. As its dimen- 
sions became less, its velocity of rotation became greater, ac- 
cording to the laws of Mechanics: since any particle moving in 
a circle of any radius with a certain linear velocity w r ould, as it 
approached the centre, move in a smaller circle with nearly the 
same linear velocity, and w 7 ould therefore have a greater angular 
velocity. Finally, the centrifugal force generated by this increased 
velocity at the surface of the equator of the mass exceeded the 
attraction towards the centre, and a nebulous zone was detached, 
which revolved independently of the interior mass, just as the 
rings of Saturn have been seen to revolve about that planet. This 



NEBULAR HYPOTHESIS. 185 

zone, by concentration at certain points within itself, broke up 
into separate masses; and these masses, either from slight differ- 
ences of velocity or from the preponderating attraction of some 
fraction larger than the others, eventually formed one body, re- 
volving about the central mass. And, furthermore, as these 
separate masses came together, a motion of rotation was com- 
municated to the combined mass, just as a whirlpool or an eddy is 
formed when two streams of water meet; and this rotating mass, 
condensing and contracting in its turn, threw off from itself a 
second zone, which underwent all the changes above described. 
Thus were formed a planet and its satellite, each revolving about 
its primary in the direction of that primary's axial rotation: and 
by a continuation of the process the whole system of planets and 
satellites was evolved. 

230. Necessary Conditions. — It is a necessary condition of the 
truth of this hypothesis, that the planets shall revolve (as they do 
revolve) about the sun in the same direction in which it rotates. 
It is also necessary that each satellite or system of satellites shall 
revolve about its primary in the same direction in which that 
primary rotates. It is not, however, absolutely necessary that 
the outer planets shall rotate in the same direction in which they 
revolve; although such a coincidence might be expected, since 
the revolution of the outer particles from which a planet was 
formed would be more rapid than that of those which were 
nearer to the sun. 

If we assume this hypothesis to be true, the rings of Saturn are 
to be considered as rings which did not form satellites after they 
were thrown off from the planet; while in the case of the minor 
planets the ring broke up into separate masses, which have con- 
tinued to revolve in independent orbits about the sun. 

231. Experiment in Support of the Hypothesis. — The possible 
truth of the nebular hypothesis is supported by an ingenious ex- 
periment devised by M. Plateau.* A mass of olive-oil was im- 
mersed in a mixture of alcohol and water, the density of the mix- 
ture being made exactly equal to that of the oil. In this way 

* See Annates de Chimie, vol. xxx. (1850). The experiment is also de- 
scribed in Carpenter's Mechanical Philosophy, <£c., one of the volumes of 
Bohn's Scientific Lihrarv (London). 



186 NEBULAR HYPOTHESIS. 

the mass of oil was practically withdrawn from the influence of 
gravitation. When made to rotate, the mass assumed a sphe- 
roidal form, and finally, when the velocity of rotation was suffi- 
ciently great, a ring of matter was thrown off in the equatorial 
region. This ring subsequently broke up into independent 
masses, each of which assumed a globular form, rotated on an 
axis of its own, and continued to revolve about the central mass : 
thus presenting precisely the successive phenomena which are 
assumed in the nebular hypothesis to have occurred in the for- 
mation of the solar system. 

232. The truth of the nebular hypothesis is by no means uni- 
versally admitted by astronomers and other scientific men ; and 
it is difficult to say what is the predominant belief about it at 
the present time. The high scientific reputation of those who 
originated it, and of those who have since supported it, is suffi- 
cient justification for giving it a place in this treatise; but it 
must not be forgotten that its truth is still very emphatically an 
open question, and that many great minds are numbered w T ith 
its opponents. 

Sir William Herschel w T as led to the adoption of the nebular 
theory by his examination of that class of celestial bodies called 
nebulae, some of which presented in his day, and present now, 
the appearance of masses of nebulous matter. Recent spectro- 
scopic examinations of some of these nebulae (Art. 286) go to 
show that they are really what they seem to be, masses of incan- 
descent vapor; and this discovery gives a new interest to the ne- 
bular hypothesis. Mr. Lockyer, in his Elementary Lessons in 
Astronomy, says that "it may take long years to prove or dis- 
prove this hypothesis; but it is certain that the tendency of 
recent observations is to show its correctness." 

A statement of "Kirkwood's Law/' which may have some 
bearing on the nebular hypothesis, will be found in the Appendix. 



PLATE III. 






COMETS, 




1. BIELA'S COMET 

2, ENCKE'S COMET. 




3 
4 



GREAT COMET OF 1861, 
DONATES COMET, 1858. 



COMETS. 1^7 



CHAPTER XIII. 

COMETS AND METEORIC BODIES. 
COMETS. 

233. General Description of ComeU. — A comet is a body of 
nebulous appearance and irregular shape, revolving in an orbit 
about the sun. Comets have usually been considered to con- 
sist for the most part of nebulous matter; but the theory has 
lately been advanced that they are collections of minute meteoric 
bodies. 

Comets differ widely from each other in appearance, and no 
description of them can be given to which there will not be 
many exceptions. Generally speaking, a comet consists of three 
parts: the nucleus, the coma, and the tail. The nucleus and the 
coma together form the head. The nucleus is a bright point, 
like a star or a planet, which may be either a solid mass, or a mass 
of nebulous matter of a density greater than that of the rest of 
the comet. The diameter of the nucleus varies considerably in 
different comets: that of the comet of 1845 (in)* was about 
8000 miles, while that of the comet of 1806 was only 30 
miles. The average value is not over 500 miles: and in many 
comets no nucleus whatever is perceptible. 

The coma is a mass of cloud-like matter, more or less nearly 
globular in form, which surrounds the nucleus. The nucleus, 
however, as a general thing, is not situated at the centre of the 
coma, but lies towards that margin which is the nearer to the sun. 
The diameter of the coma is different in different comets: that 
of the comet of 1847 (v) was only 18,000 miles, while that of the 
comet of 1811 (i) was over 1,000,000 miles. Usually, however, it 

* The number (in) means that this was the third comet which appeared 
in the course of the year. 



188 COMETS. 

is less than 100,000 miles. It is frequently noticed that the coma 
decreases in apparent diameter as the comet approaches the sun, 
and increases as the comet recedes from it. On the supposition 
that the coma consists of vaporous matter, this phenomenon is 
explained by the assumption that the intense heat to which the 
comet is subjected as it approaches the sun is sufficient to rarefy 
this vaporous matter to such an extent that some of it becomes 
invisible. 

The tail is a train of cloud-like matter attached to the head, 
which usually lies in a direction nearly opposite to that in which 
the sun lies from the head. The tail is usually very small when 
the comet first appears, and sometimes is not even perceptible. 
As the comet approaches the sun, the length of the tail increases, 
and sometimes becomes enormous. In the comet of 1811 (i), for 
instance, the length of the tail was 100,000,000 miles; and in 
that of 1843 (i) it was 200,000,000 miles. 

The angular length of the tail depends not only on its abso- 
lute length, but also on its distance from the earth, and on the 
direction in which the axis of the tail lies. There are six comets 
on record of which the tails subtended angles of over 90° ; and 
one of these, that of 1861 (n), had a tail of 104° in length, as 
observed at some places. 

234. Diversity of Appearance. — The description above given 
may be considered to apply to comets taken as a class; but, as 
already remarked, important exceptions are often noticed in 
individual comets. Indeed, it is hardly possible to compare any 
two comets without finding marked points of difference in them. 
Some comets are not visible at all, except by the aid of powerful 
telescopes, and are hence called telescopic comets ; while others, 
again, are so conspicuous as to be visible to the naked eye in full 
daylight. Some comets have more than one tail ; the comet of 
1823, for instance, had a tail turned towards the sun, in addition 
to the usual one turned from it. The comet of 1744 is reported 
to have had six tails, spread out like an immense fan, through 
an angle of 117°; but the truth of the record is not above 
suspicion. 

Not only do comets differ thus widely from each other in ap- 
pearance, but even the same comet changes its appearance from 



™ L °* MS COMET 

^P.dity^nthe case of the Great S TT* ^ m "veJIou s 
was estimated to be about ? OoToo ^ W ' the "— 
COme had P^sed its perihelion Th * " da * after the 

record of a comet's having separate, H 7 S ° me "^^ « 
Th« n- asserted in the Greek record a ""° ***»* COm ^ 
w 3/0 b.c: and Biela's oorw COmet wllic h appeared 

of t-is kind. This comet" oT^ ? indubUab]e ES 
-as determined to be a com " ^ ^ ° ^ a " d 1832 > and 
Its rpt„,n ; n t^oq tomet ,VJ th a period of nearlv ««,.<, 

and then presentafcnat.oW,^ Tf ^J~? S(n eo years, 
moving side by side, at a distance apart of ovei-IW^s ,« 
235 Tie Tm7.— The general form of the tail is that of a trun- 
cated cone, the larger base being at the extremity of the tail It 
is noticed that the tail is always brighter near the borders than 
along the middle, from which it is inferred that it is hollow: 
since only on such a supposition would the line of sight pass 
through more luminous matter when directed to the edges than 
M-hen directed to the middle. With regard to the formation of 
the tail, the most generally accepted theory seems to be that the 
matter of which the nucleus is composed is excited and dilated 
by the action of the sun's rays, as the comet approaches the sun 
and that particles of vaporous matter are thrown off from it ; and 
that these particles are driven to the rear by some repulsive force 
exerted by the sun, and thus form the tail. What this repulsive 
force exerted by the sun is, has not yet been determined; but the 
general situation which the tail of a comet has with reference to 
the sun seems to justify the inference that some such force does 
exist. Nor has it yet been determined what is the force which 
originally detaches these vaporous particles from the nucleus: it 
may be the same repelling force which drives them to the rear, it 
may be a force generated in the nucleus itself, or it may be a 
combination of both these forces. If we adopt the theory of the 
meteoric structure of these bodies, the tail is to be considered as a 
cloud of minute particles of matter, held together by their rau- 



190 



TAIL OF THE COMET. 




tual attraction, or by the attraction exerted upon them by the 
denser mass which constitutes the head. 

236. Curvature of the Tail. — The tail of a comet is usually not 
straight, but is concave towards that part of space which the 
comet is leaving. If we assume the existence of a solar repul- 
sive force, similar to that mentioned in the preceding article 
this peculiarity of shape may be thus explained. In Fig. 72, let 
S be the sun, and GCI) 
a portion of the orbit of a 
comet. When the nucleus 
is at A, let a particle be 
driven from it in the di- 
rection SA, with a force 
'sufficient to carry it to L 
in the time in which the 
nucleus moves from A to C. 
When the nucleus reaches 
(7, this particle, still retain- 
ing the motion which it had in common with the nucleus, wall be 
found at some point M. In the same way a particle driven from 
the nucleus when it is at £ will be found at some point K, when 
the nucleus reaches C: and, in general, when the nucleus is at C 
the tail will not lie in the direction SN 9 but in the direction of 
the curve CKM, as shown in the figure. 

237. Elements of a Comet's Orbit. — A comet is identified at its 
successive returns, not by its appearance, which is liable, as we 
have already seen, to serious changes, but by the elements of its 
orbit. In consequence of the comparative ease with which the 
elements of a parabola can be calculated, astronomers are in the 
habit of using that curve to represent at first the approximate 
form of a comet's orbit. The elements of a parabolic orbit are 
five in number, and are as follows : — 

(1.) The inclination of the orbit to the plane of the ecliptic: 
(2.) The longitude of the ascending node: 
(3.) The longitude of the perihelion: 
(4.) The time at which the comet passes its perihelion: 
(5.) The distance of the comet from the sun at perihelion. 
Tables and formulae have been constructed by which these 



If 



NUMBER OF COMETS. 191 

elements can be computed from the results of three distinct ob- 
servations of the position of the comet: and these three observa- 
tions may all be made, if necessary, within the space of 48 hours. 
The parabolic elements having thus been obtained, the catalogues 
of comets are searched to see if these elements are similar to 
those recorded of any previous comet. As it is highly impro- 
bable that the elements of any two comets will coincide through- 
out, the presumption is a strong one, if two comets, visible at 
different times, move in the same orbit, that they are one and the 
same comet: and the more often the coincidence is repeated, the 
more nearly does the presumption approach to a demonstration. 

238. Number of Comets, and their Orbits. — The number of 
comets which have been recorded since the Christian era is 
over 730: and there are about 70 recorded as observed before 
that date. Of these 800 appearances of comets, some may un- 
doubtedly have been only reappearances of the same comet: 
and, indeed, in some cases comets have been identified with 
other comets previously observed; but this can hardly be the 
case with the majority of these bodies. Besides these comets 
thus recorded, there must have been many others so situated as 
to be above the horizon only in the day-time : and such comets 
would become visible only in case of the occurrence of a total 
solar eclipse. A coincidence of this kind is recorded by Seneca 
as having occurred 62 B.C., when a large comet was seen in close 
proximity to the sun during a solar eclipse. The improvement 
of telescopes in recent years has greatly increased the number of 
comets which become visible, and 164 have been observed within 
the last 65 years. We are justified, therefore, in concluding that 
the comets which have really come within our system since the 
Christian era are to be reckoned by thousands. Two centuries 
and more ago, Kepler made the remarkable statement that " there 
are more comets in the heavens than fishes in the ocean. " 

The orbit in which a comet moves may be either an ellipse, 
a parabola, or an hyperbola. The orbits of 293 comets have 
been subjected to mathematical investigations, and the results 
of these investigations may be thus tabulated :* 

* The list of these comets, and the facts known concerning them, are given 
in G. F. Chambers's Descriptive Astronomy (Oxford, Eng.V 



102 PERIODIC TIMES OF COMET8. 

Comets with elliptical orbits 19; 

Subsequent returns of these comets 54; 

Comets with elliptical orbits, which have not returned.. 37 ; 

•Comets with parabolic orbits 178; 

Comets with hyperbolic orbits 5. 

A comet whose orbit is either a parabola or an hyperbola will 
not return to our system ; provided, at least, that the attraction 
of other bodies does not alter the character of the orbit. It 
must be noticed, however, that some of the orbits which are 
called parabolic, may really be ellipses of an eccentricity so 
great as to render their elements undistinguishable from those 
of parabolas. In whatever conic section a comet may move, 
the sun is always at the focus. 

239. Periodic Times. — The fifty-six comets which have been 
found to move in elliptical orbits differ widely from each other in 
the length of their periods. Among the nineteen comets whose 
returns have been observed, there are seven with short periods, 
lying between three and fourteen years. There is no doubt that 
these seven comets are periodic ; but there is some uncertainty with 
regard to some others of the remaining twelve. Two elements of 
such uncertainty are the unsatisfactory character of the records 
of the observations made in the earlier ages, and the length of 
time which the periods embrace, being often several hundred 
years. Halley's comet, however, with a period of about seventy- 
five years, is unquestionably to be added to the list of periodic 
comets about which there is no doubt. There are five other 
comets, with periods not very different from that of Halley's, 
which have been discovered within the present century, and 
which have as yet made no return. With regard to the remain- 
ing comets to which elliptical orbits and periods have been 
assigned, little more can be said than that these periods embrace 
hundreds and even thousands of years. 

240. Motion of Comets in their Orbits. — The motions of comets 
in their orbits about the sun are not performed in the same 
direction, the number of those whose motion is retrograde being 
about the same as the number of those whose motion is direct. 
According to Chambers, an examination of the motions of the 
various comets shows "that with comets revolving in elliptic 



MASS OF THE COMETS. 193 

orbits there is a strong and decided tendency to direct motion. 
The same obtains with the hyperbolic orbits : with the parabolic 
orbits there is a rather large preponderance the other way ; and 
taking all the calculated comets together, the numbers are too 
nearly equal to afford any indication of the existence of a general 
law governing the direction of motion." 

The angles which the planes of the orbits make with the 
plane of the ecliptic have values ranging from 0° to 90° ; but 
" there is a decided tendency in the periodic comets to revolve 
in orbits but little inclined to the plane of the ecliptic :" and 
if we take all the comets into consideration, "we find a decided 
disposition in the orbits to congregate in and around a plane 
inclined 50° to the ecliptic." 

Owing to the great eccentricity of the orbits, some of the 
comets approach very near to the sun at the time of perihe- 
lion passage : the comet of 1843 (i), for instance, came within 
100,000 miles of the sun's surface. For the same reason the 
distances to which some of the comets with elliptical orbits recede 
from the sun are immense ; thus the comet of 1844 (n) receded 
to a distance of 400,000,000,000 miles, over 130 times the dis- 
tance of 2s eptune from the sun. The velocity of the comets at 
perihelion is sometimes enormous ; this same comet of 1843 swung 
about the sun through an arc of 180° in only two hours, and 
moved with the velocity of 350 miles a second. 

241. Mass and Density of the Comets. — The minuteness of the 
mass of the comets is proved by the fact that they exert no per- 
ceptible influence on the motions of the planets or the satellites, 
although they sometimes pass very near to them. Thus Lexell's 
comet, 1770 (i), in its advance towards the sun, became entan- 
gled with the satellites of Jupiter, and remained near them for 
five months, without sensibly affecting their motions. The effect 
of Jupiter's attraction on the comet, however, was very striking. 
The comet had a period of about five years, and yet it never 
appeared after 1770. It was found, by computation, that at 
its first return after that date it was so situated as not to become 
visible; and that in 1779, before its second return, it came 
nearer to Jupiter than Jupiter's fourth satellite : and the pre- 
sumption is that its orbit was so changed and enlarged that the 



194 LIGHT OF THE COMETS. 

comet no longer comes near enough to the earth to become visible. 
This same comet came within about 1,400,000 miles of the 
earth in 1770: near enough, had its mass been equal to that 
of the earth, to increase the length of the year by nearly three 
hours ; but no sensible effect was produced. 

The mass, then, of the comets being so small, and their 
volume so large, the density of the matter of which they are 
composed must be exceedingly rare. Indeed, it must be vastly 
more rare than that of the lightest gas or vapor of which we 
have any knowledge: for stars of the smallest magnitude are 
distinctly seen, and usually, too, with no perceptible diminution 
of brightness, through all parts of the comets excepting perhaps 
the nucleus ; and this too in cases where the volume of nebulous 
matter has a diameter of 50,000 or 100,000 miles. 

242. Light of the Comets, — The question w 7 hether or not 
comets shine by their own light does not seem to be satisfac- 
torily decided. The existence of phases would of course prove 
that they shine by reflected light ; but although in one or two 
cases the statement has been made that phases have been 
detected, the truth of the statement has in no case been univer- 
sally accepted. Undoubtedly the distance of some comets is 
so great that phases might exist and still escape observation, 
as in the case of the superior planets ; but, on the other hand, 
some comets come so near to the earth that there seems to be 
no good reason why phases, if any exist, should not be noticed. 
Observations upon the light of the comets have been made, 
both with the polariscope and with the spectroscope. The ob- 
servations made with the polariscope seem to establish the fact 
that the comets shine, partly at all events, by the reflected 
light of the sun : as, for instance, the observations of Airy and 
others on Donati's comet in 1858. Mr. Huggins, of England, to 
whom we owe so many interesting discoveries made with the 
spectroscope, has recently examined the light of several comets 
with that instrument. In Brorsen's comet he found that the 
nucleus and part of the coma shone by their own light. In 
TempeFs comet the nucleus shone by its ow T n light, and the 
coma by the reflected rays of the sun. In some comets the 
light from the nucleus resembled that which comes from the 



PERIODIC COMETS. 



195 



gaseous nebulae. In one comet there was a remarkable resem- 
blance in the spectrum produced by its light to that produced 
by carbon, not only in the position of the bands, but in charac- 
ter and relative brightness. 

Altogether, the question as to the light of the comets may 
fairly be regarded as still an open one, to be decided, perhaps, 
by future observations. 

PERIODIC COMETS. 

243. It has already been stated, in Art. 239, that there are at 
least eight comets which are undoubtedly periodic, seven of these 
being comets with short periods, and the eighth being Hal ley's 
comet. The following table contains a list of these comets. 



NAME. 


PERIOD IN 
YEARS. 


NUMBER OF 

APPEARANCES 

OBSERVED. 


LAST AP- 
PEARANCE. 


Encke, 

Winnecke or Pons, 

Brorsen, 

Biela, 

D' Arrest, 

Faye, 

Mechain or Tuttle, 

Hal ley, 


3.3 
5.5 
5.6 
6.6 
6.6 
7.4 

13.7 

76. 


20 
3 
3 
6 
o 

4 
3 
5 


1871 

1869 
1868 
1852 
1857 
1865 
1871 
1835 



ENCKE S COMET. 

244. On November 2fi, 1818, a small and ill-defined telescopic 
comet was discovered in the constellation Pegasus, by the 
astronomer Pons, at Marseilles. It remained visible for seven 
weeks, and many observations w T ere made upon it. Professor 
Encke, of Berlin, finding that the elements of the orbit did not 
agree with those of a parabola, determined to subject them to a 
rigorous investigation, according to the method proposed by 



196 PERIODIC COMETS. 

Gauss. This investigation showed that the orbit was elliptical, 
and that the period of the comet w r as about 3i years. He fur- 
ther identified the comet with the comets of 1786 (i), 1795, and 
1805, and predicted that it would return to perihelion on May 
24, 1822, after being retarded about nine days by the influence 
of Jupiter. 

" So completely were these calculations fulfilled, that astrono- 
mers universally attached the name of ' Encke' to the comet of 
1819, not only as an acknowledgment of his diligence and success 
in the performance of some of the most intricate and laborious 
computations that occur in practical astronomy, but also to mark 
the epoch of the first detection of a comet of short period ; — one 
of no ordinary importance in this department of science." 

The comet has since been observed at every reappearance, the 
appearance in 1871 being the twentieth on record. In 1835, it 
passed so near to the planet Mercury as to show conclusively that 
the generally received value of that planet's mass must be far 
too great : since the planet exerted no perceptible influence on 
the comet's orbit. 

The comet is sometimes visible to the naked eye. It usually 
appears to have no tail ; but in 1848 it had two, one about 1° in 
length, turned from the sun, and the other of a less length and 
turned towards it. At perihelion the comet passes w T ithin the 
orbit of Mercury: while at aphelion its distance from the sun is 
nearly equal to that of Jupiter. 

One very curious feature in connection w T ith this comet is that 
its period is steadily diminishing, by an amount of about 2 J 
hours in every revolution, the period having been nearly 1213 
days in 1789-92, and only about 1210 days in 1862-65. Encke's 
own theory to account for this diminution is that the space 
through which the comet moves is filled with some extremely 
rare medium, too rare to obstruct the motions of the planets, but 
dense enough to offer sensible resistance to the progress of the 
comets. The effect of this diminution of velocity is to diminish 
the comet's centrifugal force, so that the comet is drawn nearer 
to the sun, and its orbit becomes smaller. But as the orbit 
becomes less, the angular velocity of the comet is increased, and 
its period of revolution is decreased 



PERIODIC COMETS. 197 

This theory of Encke's concerning the existence of a resisting 
medium is by no means universally accepted by astronomers. 

winnecke's or pons's comet. 

245. The second comet in the list was discovered by Pons, on 
June 12, 1819. Professor Encke assigned to it a period of bi 
years, but the comet was not seen again until March 8, 1858, 
when it w 7 as detected by Winnecke, at Bonn. He was at first 
inclined to consider it a new comet, but soon identified it with 
the one previously discovered by Pons. Its distance from the 
sun at perihelion is about 70,000,000 miles, and its distance at 
aphelion about 520,000,000 miles. It again appeared in 1869. 

BRORSEX's COMET. 

246. This comet was discovered by M. Brorsen, at Kiel, on 
February 26, 1846. The orbit was found to be elliptical, with a 
period of about 5 i years, and its return to perihelion was fixed 
for September, 1851 ; but its position at that time was so un- 
favorable for observation that it was not detected. It was seen 
at its next return to perihelion, on March 29, 1857. It again 
escaped detection in 1862, but was seen in this country on May 
11, 1868. 

Its perihelion distance is 60,000,000 miles, and its aphelion 
distance, 530,000,000 miles. 

biela's comet. 

247. This comet was discovered by M. Biela, an Austrian 
officer, at Josephstadt, Bohemia, on February 27, 1826. It was 
observed for nearly two months, and was identified with comets 
which had previously been seen in 1772 and 1805. 

Its next return to perihelion was fixed for November 27, 1832 ; 
and the comet passed perihelion within twelve hours of that time. 
On October 29, 1832, it passed within 20,000 miles of the earth's 
orbit : but the earth did not reach that point of its orbit until a 
month afterwards. No little alarm w r as created, however, outside 
of the scientific world, when it became generally known how 
near to the earth's orbit the comet would approach. 

At its return in 1839 it was not observed, owing to its close proxi- 



198 PERIODIC COMETS. 

mity to the sun. It was again detected on November 28, 1845, 
and by the end of the year it was found to have separated into 
two parts, and to present the extraordinary appearance of two 
comets, moving side by side, at a distance apart of over 150,000 
miles. It again returned in 1852, and presented the same ap- 
pearance; but the distance between the parts had increased to 
over one million of miles. Since that time the comet has never 
been seen. 

Two theories have been advanced to account for this singular 
separation. One is that the division may have been the result 
of some internal repulsive force, similar to that which forms 
the tails of comets ; the other is that it may have been the result 
of collision with some asteroid. At perihelion the comet passes 
within the orbit of the earth, and at aphelion it passes beyond 
that of Jupiter. 

d'arrest's comet. 

248. This comet was discovered by Dr. D' Arrest, at Leipsic, 
on June 27, 1851. It remained in sight for about three months, 
and its period was determined to be about 6i years. Its return 
in November, 1857, was accordingly predicted, and the predic- 
tion was verified; although, owing to the comet's great southern 
declination, it was only observed at the Cape of Good Hope. 
The unfavorable situation of the comet in 1864 prevented its 
being seen on its return in that year. 

Its perihelion distance is about 100,000,000 miles, and its 
aphelion distance more than 500,000,000 miles. 

faye's comet. 

249. This comet was discovered by M. Faye, at the Paris 
Observatory, on November 22, 1843. It had a bright nucleus and 
a short tail, but was not visible to the naked eye. The elements 
of its orbit were investigated by Le Verrier, who predicted that 
it would return to perihelion on April 3, 1851 ; and it returned 
within about a day of the time predicted. It has since made 
two returns: one in 1858, the other in 1865. The dimensions 
of its orbit are nearly the same as those of D'Arrest's comet. 



PERIODIC COMETS. 199 

MECHAIx's OR TUTTLE's COMET. 

250. This comet was discovered by Mechain, at Paris, on 
January 9, 1790. Its period was calculated to be less than 14 
years; but the comet was not seen again until January 4, 1858, 
when it was detected by Mr. H. P. Tuttle, at the Harvard Col- 
lege Observatory. Its third appearance was in 1871. 

halley's comet. 

251. In the latter part of the seventeenth century, Sir Isaac 
Newton published his Principia. In that great work he as- 
sumed that the comets were analogous to the planets in their 
revolutions about the sun, although no periodic comet had then 
been discovered. He explained the methods of investigating 
the orbits of the comets, and invited astronomers to apply these 
methods to the various comets which had been observed. Hal- 
ley, a young English astronomer, and afterwards the second 
Astronomer Royal, after a careful investigation, identified the 
comet of 1682 with comets which had appeared in 1531 and 
1607: the period of the comet being about 75i years. The 
fact that the interval of time between the first and the second 
of these appearances was not exactly equal to that between the 
second and the third seemed at first to offer some difficulty; 
but Halley, "with a degree of sagacity which, considering the 
state of knowledge at the time, cannot fail to excite unqualified 
admiration," advanced the theory that the attractions of the 
planets would exert some influence on the orbits of the comets. 
Having thus decided that this comet was a periodic comet, Hal- 
ley predicted the return of the comet about the beginning of 
the year 1759 ; and the comet passed its perihelion on March 12, 
in that year. The comet again appeared in 1835, and its 
next appearance will be in 1912. 

The comet is a very conspicuous one, with a tail sometimes 
30° in length and sometimes 50°. The comet has been traced 
back through the astronomical records, with more or less cer- 
tainty, to 11 B.C., the number of appearances being about seven- 
teen. It is not impossible that it was this comet which appeared 
in 1066, when it is recorded that a large comet excited dread 



200 REMARKABLE COMETS. 

throughout Europe, and was in England considered to presage 
the success of the Norman invasion. It is also probably iden- 
tical with the comet of 1456, which had a splendid tail 60° in 
length. 

Halley's comet at perihelion is nearer to the sun than Venus, 
while at aphelion it recedes beyond the orbit of Neptune. 

REMARKABLE COMETS OF THE PRESENT CENTURY. 
THE GREAT COMET OF 1811. 

252. The comet of 1811 (i) was discovered on March 26, 1811, 
and was visible about seventeen months. It was very conspi- 
cuous in the autumn of 1811, remaining visible throughout the 
night for several weeks. Sir William Herschel states that the 
nucleus was well defined, with a diameter of about 428 miles; 
that it was of a ruddy hue, while the surrounding nebulous matter 
had a bluish-green tinge. The tail was about 25° in length, 
and 6° in breadth. Its aphelion distance from the sun is 14 
times that of Neptune, and its period, according to Argelander, 
is 3065 years, with an uncertainty of 43 years. 

THE GREAT COMET OF 1843. 

253. The comet of 1843 (i) was first seen in the southern 
hemisphere in February, and became visible in the northern 
hemisphere the next month. It was decidedly the most won- 
derful comet of the present century. Its nucleus and coma 
shone with great splendor, and its tail was a luminous train of 
about 60° in length. On the day after its perihelion passage, 
and when only 4° distant from the sun, it was seen in broad 
daylight in some parts of New England, and its distance from 
the sun was measured with a sextant. It is described as having 
been at that time as well defined, in both nucleus and tail, as the 
moon is on a clear day. The comet is remarkable for its small 
perihelion distance, which was only about 540,000 miles; so that 
the comet came within 100,000 miles of the sun's surface. The 
intensity of the heat to which the comet must then have been 
subjected is almost inconceivable. Since 540,000 miles is about 
T 4flth of the distance of the earth from the sun, and the inten- 



REMARKABLE COMETS. 201 

slty of heat varies inversely as the square of the distance, the 
heat to which the comet was subjected must have been about 
29,000 times as intense as the heat which prevails at the earth's 
surface: a heat nearly twenty times that required, as shown by 
experiments with powerful lenses, to melt agate and carnelian. 
For some days after this, the tail had a fiery red appearance; and 
its enormous length of over 200,000,000 miles, and the mar- 
vellous rapidity with which it was formed, were undoubtedly the 
results of the heat which it endured. 

The rapidity with which this comet moved about the sun has 
already been noticed (Art. 240). The period has been computed 
to be about 175 years. 

DOXATl's COMET. 

254. This comet, 1858 (vi), was discovered on June 2d, by Dr. 
Donati, at Florence. It was then only discernible with a tele- 
scope, but became visible to the naked eye about the last of 
August. Indications of a tail began to be noticed about the 20th 
of August, and in a few weeks the tail assumed a noticeable 
curvature, which subsequently became one of the most interesting 
points connected with the comet. The comet passed its perihelion 
on September 29th, and was at its least distance from the earth 
on October 10th. Its tail subtended an angle of 60°, and 
had an absolute length of 51,000,000 miles. It disappeared 
from view in the northern hemisphere in October, but was seen 
in the southern hemisphere until March, 1859. 

This comet was not as large as some others of the comets, but 
it was particularly noted for the intense brilliancy of its nucleus. 
The nebulosity surrounding the nucleus w T as also peculiar in its 
appearance. It consisted of seven luminous envelopes, parabolic 
in form, and separated from each other by spaces comparatively 
dark. These envelopes were detached in succession from the comet's 
nucleus, at intervals of from four to seven days. They receded 
from the nucleus with the daily rate of about 1000 miles. Per- 
fectly straight rays of light, or "secondary tails," were also seen. 

The comet has a period of about 2000 years. A magnificent 
memoir of this comet, by Professor G. P. Bond, is contained in 
the second volume of the Annals of the Harvard Observatory. 



202 METEORIC BODIES. 

THE GREAT COMET OF 18G1. 

255. This comet, the second of the year, was discovered in the 
southern hemisphere on May 13th, but was not seen in England 
until June 29th, about two weeks after its perihelion passage. 
The nucleus was round and unusually bright, and the tail at one 
time attained the length of over 100°. The comet remained in 
sight for about a year. 

"In a letter published at the time in one of the London papers, 
Mr. Hind, an English astronomer, stated that he thought it not 
only possible, but even probable, that in the course of Sunday, 
June 30th, the earth passed through the tail of the comet, at a 
distance of perhaps two-thirds of its length from the nucleus." 
Mr. Hind also stated that on Sunday evening there was noticed, 
by both himself and others, a peculiar illumination in the sky, 
like an auroral glare; and a similar phenomenon seems to have 
been noticed outside of London. 

According to the observations of Father Secchi, the light of 
the tail, and that of the rays near the nucleus, presented evidences 
of polarization, while the nucleus itself at first presented no 
evidences whatever; afterwards, however, the nucleus presented 
decided indications of polarization. Secchi states that he thinks 
this "a fact of great importance, as it seems that the nucleus on 
the former days shone by its own light, perhaps by reason of the 
incandescence to which it had been brought by its close proximity 
to the sun." 

METEORIC BODIES. 

256. Under the general head of meteors are included three 
classes of bodies : — 1st. The ordinary shooting stars, some of which 
can be seen rushing across the heavens on almost any clear night; 
2d. Detonating meteors, which are shooting stars, commonly of an 
unusual size, whose disappearance is followed by a sound like that 
of an explosion ; 3d. Aerolites, which, after the flash and the ex- 
plosion with which they are generally accompanied, are precipi- 
tated to the earth in showers of stones and metallic substances. 
It is only within recent years that the decided attention of astro- 
nomers has been directed to these bodies, and comparatively 



SHOOTING STARS. 



203 



little is known with certainty about them ; but the general belief 
is that they are all essentially of the same nature, differing from 
each other rather in size and density than in other more impor- 
tant respects. 

257. Shooting Stars. — Scarcely a clear night passes during 
which shooting stars are not seen. The average number of those 
which can be seen at any place by one observer, on a cloudless, 
moonless night, is estimated to be about six an hour. There is, 
however, an hourly variation in the number observed, the mini- 
mum occurring about 6 p.m., and the maximum about 6 a.m. 
According to a French writer on this subject, the mean number 
of meteors observed is given in the following table : — 



HOURS 


7-8 

P.M. 


9-10 


11-12 


1-2 

A.M. 


3-4 


5-6 




MEAN NUMBER.. 


3.5 


4 


5 


6.4 


7.8 


8.2 



It is further estimated that the number seen at any one place 
by a number of observers sufficient to watch the whole hemi- 
sphere of the heavens is 42 an hour, on the average, or about 
1000 daily: and that the number which could be seen daily over 
the whole earth, under favorable circumstances, is more than 
8,000,000. This is the number of those large enough to be vis- 
ible to the naked eye : it is simply impossible to estimate the 
number of those which could be seen with the aid of telescopes. 

It is further noticed that there are more shooting stars observed 
in the second half of the year than in the first. At certain sea- 
sons of the year, either in consecutive years or after the lapse of 
a certain number of years, there are unusually brilliant displays 
of these meteors, which are called star shoivers. The number of 
recognized star showers now exceeds fifty; and prominent among 
them are the shower of August 9-11 and that of November 
11-13. 

Professor Harkness, of the Washington Observatory, after an 
elaborate investigation of the quantity of matter in the ordinary 
shooting star, concludes that it is not far from one grain. 

258. The November Shower. — There are several historical 



204 NOVEMBER SHOWER. 

notices of brilliant displays of meteors which occurred in the 
early centuries of the Christian era : and ten of these, occurring 
between the years 902 and 1698, took place in October or 
November. The first display, however, of which we have any 
detailed account, occurred in 1799, on the morning of the 13th 
of November, and was visible over nearly the whole of the 
western continent. Humboldt witnessed it in South America, 
and thus describes it : — " Towards the morning of the 13th we 
witnessed a most extraordinary scene of shooting meteors. 
Thousands of bodies and falling stars succeeded each other 
during four hours. Their direction was very regular, from north 
to south. From the beginning of the phenomenon there was not 
a space in the firmament equal in extent to three diameters of 
the moon which was not filled every instant with bodies or fall- 
ing stars. All the meteors left luminous traces or phosphor- 
escent bands behind them, which lasted seven or eight seconds." 
Similar showers also occurred on the same day of the month 
in the years 1831, 1832, and 1833, the last one being the most 
splendid on record. It lasted from ten o'clock on the night of 
the 12th to seven o'clock on the morning of the 13th, and was 
visible over nearly the whole of North America. The display 
reached its maximum about four a.m. An observer at Boston 
about six o'clock counted 650 shooting stars in a quarter of an hour. 
Large fireballs with luminous trains were also seen, some of 
which remained visible for several minutes. Even stationary 
masses of luminous matter are said to have been seen : and one 
in particular is mentioned as having remained for some time 
in the zenith over the Falls of Niagara, emitting radiant streams 
of light. 

The November shower was witnessed again in 1866, both in 
this country and in Europe; but the display was much more 
.brilliant in Europe. The maximum seems to have taken place 
about two a.m. on the 14th, when nearly 5000 meteors were 
counted in an hour at Greenwich. At half-past one, 124 were 
counted in one minute. The case was reversed with the shower 
of 1867, the display being more brilliant in this country than in 
Europe. The report on the shower from the United States 
Observatory at Washington states that as many as 3000 were 



HEIGHT OF METEORS. 205 

counted in one hour. The most magnificent phase seems to 
have occurred about half-past four a.m. on the 14th. Professor 
Loomis states that at New Haven about 220 a minute were 
counted at this time. Many others were undoubtedly rendered 
invisible by -the light of the moon, which was then very nearly 
full. Most of the brighter meteors left trains of phosphorescent 
light, which remained visible for several seconds, and in some 
cases for several minutes. 

In 1868, the display began somewhat before midnight on the 
13th and continued until daybreak on the 14th. Professor 
Eastman, of the Washington Observatory, says in his report, 
that " considering the number and brilliancy of the meteors, 
their magnificent trains, and the magnitude of the meteoric 
group through which the earth passed, this shower was unques- 
tionably the grandest that has ever been witnessed $t this Obser- 
vatory." Over 5000 meteors were counted, and it was estimated 
that at five A. m. on the 14th the number falling in the whole 
heavens was about 2500 an hour. Several very brilliant meteors 
were observed. One in particular was brighter than Jupiter. 
It was at first of a deep orange color, afterwards green, and finally 
light blue. It left a train of 7° in length, which passed through 
the same changes of color, and remained visible for half an 
hour. The paths of 90 meteors were traced upon a chart, and 
were found in nearly every instance to start from a point in the 
constellation Leo. 

259. Height, &c. of the Meteors. — Concurrent observations 
were made at Washington and Eichmond, in November, 1867, 
for the purpose of determining the parallax of the meteors, and 
thence their distance. It w T as found that they appeared at an 
average height of 75 miles, and disappeared at the height of 
55 miles. The velocity with which they moved relatively to 
the earth was 44 miles a second. Other observations have given 
nearly the same results. 

The light of the meteors is probably due to the intense heat 
generated by the resistance of the air to the progress of these 
bodies. Notwithstanding the extreme rarity of the air at the 
height of the meteors, it is still believed that the heat resulting 
from such immense velocity is sufficient to fuse any known sub- 



206 ORBITS OF METEORS. 

stance. A body moving with this velocity at the earth's surface 
would acquire a temperature of at least 3,000,000°. An exami- 
nation of the light of the meteors with the aid of the spectro- 
scope, by Mr. A. Herschel, showed that some of the meteors 
were solid bodies in a state of ignition, but that most of them 
were gaseous. 

260. Orbits of the Meteors. — It is noticed that the November 
meteors, or at all events the great majority of them, seem to 
come from the same point in the heavens, — a point in the con- 
stellation Leo. So also the August meteors come from a point 
near the head of the constellation Perseus. Such points are 
called radiant points. Other showers have also other radiant 
points, situated in various parts of the heavens. The number 
of such points now recognized is more than 60. The paths in 
which meteors having the same radiant point move during the 
instant of time that we see them, are really parallel straight 
lines, the apparent convergence of the paths being merely the 
result of perspective; in other words, the radiant point is the 
vanishing point (Art. 16) of these parallel lines. 

Knowing the direction and the velocity with respect to the 
earth of the motion of a meteor, it is easy to compute the same 
elements of its motion with reference to the sun. The results of 
such computation, together with the existence of the radiant 
points and the periodic recurrence of showers, have led to the 
theory that the November meteors are collected in a ring, or in 
several rings, or possibly in a series of clusters or groups, which 
revolve about the sun; and that the showers occur when the 
earth encounters these rings or groups. The theory of one ring is 
exemplified in Fig. 73.* Let AB CD represent the orbit of the 
earth, and AGBE a ring of meteors revolving about the sun. 
If A and B are the points at which the earth enters the ring, 
there will be displays of meteors, when the earth is at these 
points, similar to the August and the November shower. Unless 
the plane of the ring coincides with the plane of the ecliptic, 
there will be no showers at any other points of the earth's orbit. 
If we imagine the ring to be broken, or to be of unequal 

* Phipson's Meteors, Aerolites, and Falling Stars. London, 1867. 



OUBITS OF METEORS. 



207 



thickness, or to consist of a series of groups, we may account for 
the irregularities which exist in the annual showers ; and by 




Fig. 73. 



supposing the ring to have a period of about 33} years, we may 
account for the extraordinary displays of meteors which happen 
in about that interval of time. The duration of a shower, and the 
known velocity of the earth in its orbit, enable us to obtain an 
approximate value of the breadth of the ring. Professor East- 
man estimates that the breadth of that portion of the ring through 
which the earth passed in November, 1868, could not have 
been less than 115,000 miles. The breadth of the stream in 
1867 was less than this, but more densely packed with meteors. 

Le Verrier, a French astronomer, has computed the elements 
of the orbit of the November meteors. He finds the major 
semi-axis to be 10.34, the perihelion distance 0.989 (the radius 
of the earth's orbit being unity), and the eccentricity 0.9044. 
This would carry the aphelion beyond the orbit of Uranus, if 
both orbits w T ere projected upon the plane of the ecliptic. 

According to Professor Loomis, the relative situations of the 
orbit of the November meteors and the orbits of the earth and 
the other planets, are represented in Fig. 74; the orbit of the 
meteors being a very eccentric ellipse, the aphelion of which 
lies beyond the orbit of Uranus, and the period of the me- 
teors being 33} years. According to the same authority, the 
August meteors revolve in a similar but much more eccentric 
ellipse, of which the aphelion lies far beyond the orbit of Nep- 
tune. The theory has also been advanced that meteors (or, at 
all events, some of them) are to be regarded as satellites of 



208 



DETONATING METEOKS. 



the earth rather than of the sun. On this subject Sir John 
Herschel says, in his Outlines of Astronomy : — "It is by no means 

inconceivable that the earth, 
approaching to such as dif- 
fer but little from it in direc- 
tion and velocity, may have 
attached them to it as per- 
manent satellites, and of these 
there may be some so large 
as to shine by reflected light, 
and to become visible for a 
brief moment ; suffering, after 
that, extinction by plunging 
into the earth's shadow." 

261. Detonating Meteors. — 
The height and the velocity 
of these bodies are not essen- 
tially different from those of 
the ordinary shooting stars. 
They are, however, generally 
of an unusual brilliancy, and 
their appearance is followed 
by an explosion, or a series 
of explosions, the intensity 
of which is sometimes terrific. 
Records of more than eight 
hundred detonating meteors 
are to be found in scientific journals. The phenomena connected 
with the appearance of these bodies are, however, so nearly iden- 
tical in character, that one instance may suffice to exemplify all. 
"On the 2d of August, 1860, about 10 p.m., a magnificent fire- 
ball was seen throughout the whole region from Pittsburg to 
New Orleans, and from Charleston to St. Louis, an area of 900 
miles in diameter. Several observers described it as equal in 
size to the full moon, and just before its disappearance it broke 
into several fragments. A few minutes afte** the flash of the 
meteor there was heard throughout several counties of Ken- 
tucky and Tennessee a tremendous explosion, like the sound of 




Fig. 74. 



AEROLITES. 209 

distant cannon. Immediately another noise was heard, not 
quite so loud, and the sounds were re-echoed with the prolonged 
roar of thunder. From a comparison of a large number of 
observations, it has been computed that this meteor first became 
visible over Northeastern Georgia, about 82 miles above the 
earth's surface, and that it exploded over the southern boundary 
line of Kentucky, at an elevation of 28 miles. The length of 
its visible path was about 240 miles, and its time of flight eight 
seconds : showing a velocity relative to the earth of 30 miles per 
second. It is hence computed that its velocity relative to the 
sun was 24 miles per second." 

The explosions are probably due to the sudden compression 
and shocks to which the air is subjected as the meteor rushes 
through it, as happens when a gun is fired; or to the rushing 
of the air into the vacuum which the body creates in its rear. 
The appearance of these bodies is so sudden, and their velocity 
so great, that it is almost impossible to obtain any definite value 
of their magnitude. The diameters of some of them are stated 
to have been several thousand feet in length, but the estimate 
must be taken w 7 ith considerable caution, particularly as it is 
impracticable to distinguish between the meteor itself and the 
blaze of light which surrounds it. 

262. Aerolites. — Although the ordinary shooting stars some- 
times appear to break in pieces, there is no evidence that any 
part of them falls to the earth. But occasionally solid masses 
of stone or of metallic substances do fall to the earth, their fall 
being usually preceded by the flash and the discharge of a deto- 
nating meteor. There is no doubt whatever about the authen- 
ticity of most of these cases, and the record of them extends 
far back into ancient history. A fall of meteoric stones near 
Rome, 650 years before Christ, is mentioned by the historian 
Livy; and a large block of stone is said to have fallen in 
Thrace, near what is now called the Strait of Dardanelles, 465 
years before Christ. The entire number of aerolites of which 
we have any determinate knowledge is more than 400 ; and more 
than twenty falls of aerolites have occurred in the United States 
since the beginning of the present century. The British Mu- 
seum contains a large collection of aerolites, one of which 

H 



210 AEROLITES. 

weighs 8287 pounds ; and many other similar specimens are to 
be found in the cabinets of colleges and museums, both in this 
country and in Europe. The following are instances of falls 
which have occurred since 1800. 

In 1807, on the morning of December 14th, a brilliant meteor, 
with an apparent diameter equal to about one-half of that of 
the moon, was seen moving over the town of Weston, in the 
southwestern part of Connecticut. After its disappearance, 
three loud explosions were heard, followed by a continuous 
rumbling. Fragments of stone were precipitated to the earth 
within an area of a few miles in diameter. The entire weight 
of these fragments is estimated to be about 300 pounds. One 
fragment, weighing 36 pounds, is preserved in the museum 
at Yale College. The specific gravity of the aerolite was about 
3 J, and among its components were silex, oxide of iron, magne- 
sia, nickel, and sulphur. 

On the 1st of May, 1860, about noon, there was a number 
of explosions over the southeastern part of Ohio. Stones were 
seen to fall to the earth, and in some cases they penetrated the 
earth to a distance of three feet. About thirty fragments were 
found, the largest of which weighs 103 pounds, and is to be 
found in the cabinet of Marietta College. The combined weight 
of these thirty fragments is not far from 700 pounds, and the 
specific gravity and the composition are very similar to those of 
the Weston aerolite. 

Another phenomenon of this character occurred in Piedmont, 
on February 29, 1868, about the middle of the forenoon. There 
was a heavy discharge like that of artillery, followed, after a 
short interval, by a second discharge. A mass of irregular 
shape was seen in the air, enveloped in smoke, and followed by 
a long train of smoke. Other bodies, similar in appearance to 
meteors, were also seen. The analysis of the stones which fell 
showed the existence in them of the components mentioned 
above, and also of copper, manganese, and potassium. 

Other aerolites have been subjected to chemical analysis. 
Of the 65 elementary substances known, 19 at least have been 
found in aerolites, and no new elements have been discovered. 
A meteoric shower usually consists of meteoric iron and meteoric 



CONNECTION OF COMETS AND METEORS. 211 

stone ; the iron is an alloy of which the principal part is nickel, 
and which also contains cobalt, tin, copper, manganese, and 
carbon; the stone contains chiefly those minerals which are 
abundant in lava and trap-rock. The proportions in which 
these ingredients enter into the composition of different aerolites 
differ greatly : sometimes an aerolite contains 96 per cent, of 
iron, sometimes scarcely any iron at all. A substance called 
schreibersite, which is a compound of iron, nickel, and phos- 
phorus, is always found in these bodies. 

The explosion of an aerolite may be due either to the intense 
heat generated by its rapid motion, or to the pressure to which 
it is subjected by the resistance of the atmosphere. 

263. Origin of Aerolites. — Many theories have been advanced 
to explain the origin of these bodies. One theory is that they 
may be formed in the atmosphere by the aggregation of minute 
particles drawn up from the surface of the earth ; but one objec- 
tion to this theory is that it does not account for the nearly 
horizontal direction in which they move, and for the great 
velocity of their motion. A second theory, that they are thrown 
from terrestrial volcanoes, is open to the same objection. A 
third theory, that they may be ejected from the volcanoes of the 
moon, is weakened by the fact that observation shows no signs 
(or at least almost no signs) of activity in the lunar volcanoes. 
The most probable theory is that they are, like the planets and 
the comets, satellites of the sun, revolving about it in orbits 
which intersect the orbit of the earth, and that their fall to the 
earth's surface is either the direct result of their own motion, 
or is due to the resistance of the atmosphere, and the attraction 
exerted upon them by the earth. 

264. Possible Connection of Comets and Meteoric Bodies. — The 
facts which have been presented in this chapter in relation to 
comets and meteoric bodies point to one certain conclusion : — 
that space, or at least that portion of space through which the 
earth moves, must be considered to be filled w T ith a countless 
number of comparatively minute bodies, the aggregate mass of 
which cannot fail to be very great. In 1848, Dr. Mayer, of 
Germany, advanced a theory that the light and the heat of the 
sun are caused by the incessant collision of meteoric bodies with 



212 CONNECTION OF COMETS AND METEORS. 

its surface. In connection with this subject, Professor William 
Thompson states that if the earth were to fall into the sun, the 
amount of heat generated by the shock would be equal to that 
which the sun now gives out in 95 years ; and that the planet 
Jupiter, under similar circumstances, would generate an amount 
of heat equal to that given out by the sun in 32,000 years. 

There is a striking similarity between the elements of the 
orbit of Tempel's comet and those of the orbit of the Novem- 
ber shower, mentioned in Art. 260. There is also a similar 
coincidence in the orbit of the August shower and that of the 
Great Comet of 1862 ; and the opinion is gaining ground with 
astronomers, not only that each of these comets leads the group 
with the elements of whose orbit its own elements so nearly 
coincide, but also that there is a close connection, generally, 
between comets and meteoric bodies. The following statement 
of this new theory is taken from an article by Professor Simon 
Newcomb, in the North American Review of July, 1868. 

" The planetary spaces are crowded with immense numbers 
of bodies which move around the sun in all kinds of erratic 
orbits, and which are too minute to be seen with the most power- 
ful telescopes. 

" If one of these bodies is so large and firm that it passes 
through the atmosphere and reaches the earth without being 
dissipated, we have an aerolite. 

"If the body is so small or so fusible as to be dissipated in 
the upper regions of the atmosphere, we have a shooting star. 

" A crow r d of such bodies sufficiently dense to be seen in the 
sunlight constitutes a comet. 

"A group less dense will be entirely invisible, unless the 
earth happens to pass through it, when we shall have a meteoric 
shower." 

It is not impossible to conceive that the planets themselves 
may have been formed by the aggregation of these minute 
bodies, a method of formation exactly the opposite of that 
which is set forth in the nebular hypothesis. An article in 
which such an origin is suggested for the planets, the comets, 
and Saturn's rings, will be found in the North American Review 
for 1864. 



FIXED STARS. 213 



CHAPTER XIV. 

THE FIXED STARS. NEBULiE. MOTION OF THE SOLAR SYSTEM. 

265. We have now examined the motions and the orbits of 
all the known members of the solar system. We have seen that 
the planets and their satellites, besides their apparent diurnal 
motion towards the west in orbits whose planes are perpen- 
dicular to the axis of the celestial sphere, have also independent 
motions of their own in elliptical orbits, the sun being at the 
common focus of the orbits of the planets, and each planet being 
at the common focus of the orbits of its satellites. Besides these 
bodies, there is a vast number of other bodies visible in the 
heavens, the phenomena presented by which are radically differ- 
ent from those which have hitherto been noticed. Continuous 
observations have been made upon the stars from year to year, 
and even from century to century ; and it has been found that, 
after the results of these observations have been freed from the 
effects of precession and nutation (which, by shifting the position 
of the points or the planes of reference, may give the stars an 
apparent motion), the real change of position of the stars is 
extremely small. Sirius, the brightest star in the whole heavens, 
has an annual motion of 1" ; a Centauri, the brightest star in 
the southern hemisphere, has an annual motion of nearly 4" ; 61 
Cygni and e Indi, both small stars, have annual motions of 
5" and 7" respectively. In only about 30 stars, however, has 
the amount of this change of position been found to be greater 
than 1" a year ; and in the others, if any motion at all is de- 
tected, it is only that of a few seconds in a century. These 
motions are called proper motions, to distinguish them from 
those which are only apparent, and the stars are called fixed 
stars : a term which must be understood to imply, not that 
they have no motion in space, but that whatever motion they 



214 FIXED STARS. 

have makes no perceptible alteration in their position upon the 
celestial sphere. 

When a star moves obliquely to the line joining the earth and 
the star, its motion in its orbit can be resolved into two motions : 
one along the line of sight, either directly towards the earth or 
directly from it, and the other at right angles to that line. This 
latter motion will cause the star to shift its apparent position 
upon the celestial sphere, and will be the proper motion above 
described ; or, as it may be called, the transverse proper motion. 
Now, it is evident that in order to obtain the real motion of any- 
star in space we must be able to determine, not only its trans- 
verse motion, but also its motion towards or from the earth. As 
long as the detection of such a motion depended upon our ability 
to detect either an increase or a diminution of the star's bright- 
ness, its immense distance from us rendered the task a hopeless 
one ; but very recently the spectroscope has afforded us the means 
of solving this problem. The details of this method will be 
given at the end of this Chapter. 

266. The Number of the Fixed Stars. — The number of stars in 
the entire sphere which are visible to the naked eye is between 
6000 and 7000, according to the largest estimate ; but the number 
of those visible at any one time at any place is less than 3000. 
By means of telescopes, thousands and even millions of other 
stars are brought into view, in such numbers as almost to defy 
any attempt at computation. 

267. Magnitudes. — The fixed stars are classified arbitrarily by 
astronomers according to their relative brightness, the different 
classes receiving the name of magnitudes, and the first magnitude 
comprising those stars which are the brightest. Different astro- 
nomers, however, sometimes assign different magnitudes to the 
same star. According to Argelander's classification, there are 
20 stars of the first magnitude, 65 of the second, 190 of the third, 
425 of the fourth, &c, the numbers in the following magnitudes 
increasing very rapidly. It is estimated that there are at least 
20,000,000 stars in the first fourteen magnitudes. Those stars 
which are visible to the naked eye are comprised in the first six 
magnitudes ; and stars of the twentieth magnitude are detected 
with the most powerful telescopes. 



CONSTELLATIONS. 215 

There is, however, a great difference in the brightness of stars 
which belong to the same magnitude. Sirius, for instance, is 
fifteen times as bright as some other stars of the first magnitude.* 

268. Constellations. — In order to facilitate the formation of 
catalogues of the stars, they are separated into groups, called 
constellations. Ptolemy, in the second century, enumerated 48 
constellations: 21 northern, 12 zodiacal, and 15 southern. The 
twelve zodiacal constellations have the same names that the signs 
of the zodiac bear, which are given in Art. 91 ; indeed, the signs 
really took their names from the constellations. Owing, how- 
ever, to the precession of the equinoxes, the signs and the con- 
stellations no longer coincide (see Art. 119), the constellation of 
Aries being in the sign of Taurus, &c. 

Since the time of Ptolemy, about sixty other constellations 
have been added to the list. Not all of these, however, are ac- 
cepted by astronomers, and the list of those constellations which 
are generally acknowledged comprises only about 86 : 29 north- 
ern, 12 zodiacal, and 45 southern. 

269. Remarkable Constellations. — The most remarkable of the 
northern constellations is that called Ursa Major, or the Great 
Bear, often called " The Dipper" from the well-known appearance 
presented by its seven conspicuous stars. The two of these seven 

* Any one ?f the most proinijient stars may be identified when on the 
meridian, in the following manner. The right ascension of the star, given 
in the Ephemeris, is, by Art. 9, the local sidereal time of the star's transit ; 
and from this sidereal time the local mean solar time can be obtained, as 
proved in Art. 105, by subtracting from it the right ascension of the mean 
sun. When only a rough estimate is desired, the quantity given on page II. 
of each month in the Ephemeris, in the last column on the right, may be 
taken as the mean sun's right ascension. This will give the time of transit 
within four minutes. The more rigorous process is to apply to the quan- 
tity taken from page II. a correction taken from Table III. in the Appendix 
to the Ephemeris, using the longitude of the place as an argument, and 
adding the correction if the longitude is west. The result is then subtracted 
from the sidereal time as above, and the remainder is diminished by a 
correction taken from Table II., with the remainder itself as an argument. 
(See example in the latter part of the Ephemeris.) 

The star's meridian altitude is found by the method given in the note 
to Art. 181. 



216 CONSTELLATIONS. 

stars which are the most remote from the handle of the dipper 
are called the pointers, since the right line joining them will 
always, when prolonged, pass very nearly through the pole-star. 
This constellation contains fifty-three stars of the first five magni- 
tudes, including one of the first and three of the second. 

There is another "Dipper," much less conspicuous, consisting 
also of seven stars, the pole-star being at the extremity of the 
handle. These stars form a part of the constellation Ursa Minor, 
which contains, in all, twenty-three stars of the first five magni- 
tudes. The pole-star itself is of the second magnitude. 

The constellation Orion is a magnificent one, and was fan- 
cifully supposed by the ancients to bear some resemblance to a 
giant. It contains two stars of the first magnitude, four of the 
second, and thirty-one of the next three magnitudes. The three 
stars, situated nearly in a straight line, which form the giant's 
belt, are a very conspicuous part of the constellation. This 
constellation, in the northern hemisphere, bears south about 9 p.m. 
in the early part of February ; its altitude at that time at any 
place being nearly equal to the co-latitude of that place. Sirius, 
the brightest star in the heavens, is also seen at that time to the 
left of this constellation, and a little below it. 

The constellation Pegasus contains forty-three stars of the 
first five magnitudes. Four of these stars are of the second 
magnitude, and nearly form a square. This square bears south 
about 9 p.m. in the early part of October, with an altitude, in 
the northern hemisphere, about 15° greater than the co-latitude 
of the place of observation. 

The constellation Gemini, or the Twins, takes its name from 
two bright stars, nearly of the first magnitude, called Castor 
and Pollux. They are situated near each other, and bear south 
about 9 p.m. in the latter part of February, in the northern 
hemisphere, their altitude being about 30° greater than the co- 
latitude of the place of observation. 

270. Stars of the Same Constellation. — Stars of the same con- 
stellation are distinguished from each other by the letters of the 
Greek or the Roman alphabet, or by numerals. The Greek 
letters were first used by Bayer, a German astronomer, in 1604, 
who called the brightest star in a constellation a, the next 



NAMES OF STARS. 



217 



brightest,?, &c. Thus the pole-star bears the astronomical Dame of 
a Ursse Minoris, and the pointers of the dipper are called a and 
/5 Ursse Majoris. Owing, however, either to carelessness on the 
part of Bayer or to changes in the brightness of some of the stars, 
this alphabetical arrangement dees not in all cases accurately 
represent the relative brilliancy of the stars in a constellation. 

The entire number of stars now catalogued amounts to several 
hundreds of thousands. Three catalogues published by Arge- 
lander, in 1859-62, contain over 320,000 stars observed at Bonn. 
In large catalogues, the stars are usually numbered from be- 
ginning to end, in the order of their right ascensions. 

271. Stars with Special Names. — Some of the stars, particularly 
the more conspicuous ones, have special names, which were given 



*a Eridani A chernar. 

a Tauri Aldebaran. 

a Auriga? Capella. 

ft Orionis Kigel. 

*a Argus Canopus. 

a Canis Majoris Sirius. 

a Canis Minoris Procyon. 

/? Gerninorum Pollux. 

a Leonis Eegulus. 

a Virginis Spica. 

a Bootis Arcturus. 

a Scorpii Antares. 

a Lyra? Yega. 

a Aquilse Altair. 

a Piscis Australis Fomalhaut. 



to them by ancient astronomers. Instances are given in the 
accompanying table, all the stars contained in it being commonly 
considered to be of the first magnitude. About 90 stars are 
thus named, though most of these names are no longer used. 



218 CONSTITUTION OF STARS. 

All of these stars, excepting those marked with an asterisk, 
come above the horizon throughout the United States. The 
position of each, in right ascension and declination, can be 
found in the Ephemeris for any day in the year. Besides the 
stars in the preceding table, there are three others of the first 
magnitude : a. Crucis, a Centauri, and /5 Centauri. They are all 
stars of large southern declination, and do not come above the 
horizon in any part of the United States excepting the southern 
parts of Texas and Florida. 

272. Constitution and Diversity of Brightness. — The spectra 
of stars are found to contain dark lines, similar in character to 
those by which we have seen that the solar spectrum is distin- 
guished (Art. 102). These systems of lines differ from the system 
of lines in the solar spectrum, and they are also different in dif- 
ferent stars. This difference, however, consists in the absence 
of certain lines seen in the solar spectrum, and not in the pre- 
sence of new ones: nearly every line observed having its 
counterpart in the solar spectrum. The examination of these 
spectra, and the comparison of the dark lines which they con- 
tain with the bright lines found in the spectra of terrestrial 
substances, enable us to establish the presence of certain of 
these substances in the stars, precisely as was done in the case 
of the sun. In Aldebaran, for instance, the presence of sodium, 
magnesium, tellurium, calcium, antimony, iron, bismuth, and 
mercury, has been detected; in Sirius, of sodium, magnesium, 
iron, and hydrogen ; in a Orionis, of magnesium, sodium, cal- 
cium, and bismuth. 

The diversity of brightness in the stars may be due either to 
a difference in their distances from us, or to a difference in their 
actual magnitudes, or to a difference in the intrinsic splendor 
with which they shine. Probably all these causes exist ; but it 
is fair to conclude that, as a general rule, the brightest stars are 
the nearest to us. Observations made for the purpose of deter- 
mining the distances of the stars go to justify such a conclusion ; 
although they also show that the rule is not an absolute one, 
since some of the fainter stars are found to be nearer to us than 
some of the brighter ones. 

273. Distance of the Fixed Stars. — No perceptible difference 



DISTANCE OF THE STARS. 219 

is detected in the position of a star when observed at places of 
widely different latitudes. The conclusion drawn from this 
fact is, that the stars are so distant that lines drawn from any 
two points on the earth's surface to the same star are sensibly pa- 
rallel : in other words, that the stars have no geocentric parallax. 
To determine the distance of the stars, then, we must have re- 
course to their heliocentric parallax. In Fig. 75 let S be the 
sun, AE'BE the orbit of the earth, s the position of a fixed 




Fig. 75. 



star, supposed to lie in the plane of the ecliptic, and NM a por- 
tion of the celestial sphere. From s draw lines sE, sE', tan- 
gent to the earth's orbit, and also prolong them beyond s, until 
they meet the arc of the celestial sphere NM. Draw the radii 
vectores SE f and SE to the points of tangency. If we suppose the 
star to be at rest, it will lie in the direction E's, when the earth 
is at E\ and in the direction Es, when the earth is at E, and 
the motion of the earth about the sun will give the star an ap- 
parent oscillatory movement over the arc ee. The true helio- 
centric direction in which the star lies is Ss ; and the difference 
of the directions in which the star lies at any time from the sun 
and the earth is its heliocentric parallax. This difference of 
direction is evidently at its maximum when the earth is at E' or 
E; and this maximum, or the angle SsE, is called the annual 
heliocentric parallax, or simply the annual parallax. 

Numerous attempts have been made to determine the annual 
parallax of the stars, by comparing observations made when the 
earth is at E and E '. The nicest observation, however, has failed 
to detect in any star a parallax as gi*eat as 1"; and in only 12 
stars has the slightest appreciable parallax been discovered. 



220 DISTANCE OF THE STARS. 

274. The distance of the fixed stars, then, is so great that the 
radius of the earth's orbit, 92,400,000 miles, does not subtend 
an angle of even 1" at that distance. If, in the triangle SsE, 
we suppose the angle SsE to be equal to 1", we shall have, 

Ss = 92,400,000 cosecl": 
which will be found to be about nineteen trillions of miles. This 
is only the inferior limit of the distance of the stars : that is to 
say, whatever the distance may be, it cannot be less than this ; 
but how much greater it may be, particularly in the case of those 
numerous stars in which no movements whatever of parallax 
can be detected, it is impossible to calculate. 

275. Immensity of this Distance. — It is hardly possible to obtain 
a clear conception of a distance of nineteen trillions of miles. 
Perhaps the nearest approach to such a conception is made by con- 
sidering that light, moving with a velocity of 186,000 miles a 
second, and passing from the sun to the earth in a little more than 
eight minutes, would consume about 31, years in accomplishing 
such a distance: so that when we look at the brightest stars in 
the heavens, we see them, not as they are now, but as they were 
3 J years ago ; and if any one of them were to be destroyed at 
any instant, we should continue to see its image for three years 
and more after that time. 

Before such distances, the dimensions of the solar system 
shrink to the insignificance of a mere point in space. Neptune, 
the most distant of the planets, is nearly three billions of miles 
from the sun ; and yet, if Neptune and the sun could both be 
seen from the nearest fixed star, the angular distance between 
them would never be greater than about 30", which is only about 
^th of the angle which the sun subtends to us. 

276. Differential Observations. — In dealing with so small a 
quantity as the annual parallax of the stars, it is important to 
avoid all circumstances by which even the most minute errors 
may be entailed upon the observations. The apparent position 
of a star is affected, not only by parallax, but by precession, 
nutation, aberration, and the star's own proper motion. The 
laws of precession and nutation enable us to decide what amount 
of the apparent motion is due to them ; and the effect of a star's 
proper motion is also readily separated from that of parallax, 



DISTANCE OF THE STARS. 



221 



since the former changes the position of the star from year to 
year, while the latter only changes its position during the year, 
causing it to lie now on one side and now on the other of its 
true position, but giving it no annual progressive motion. But 
it is not so easy to separate the effects of aberration and paral- 
lax. Aberration, as we have already seen (Art. 125), causes a 
star to describe a circle, an ellipse, or an arc, about its true po- 
sition as a centre, according to its situation with reference to 
the plane of the ecliptic ; and it is easy to see that the paral- 
lactic movement of a star is of precisely the same character. 
Thus we have already seen, in Fig. 75, that a star situated in 
the plane of the ecliptic will oscillate by parallax over the arc 
ee ; and if the star is not in the plane of the ecliptic, lines 
drawn from all points of the earth's orbit to the star, and thence 
prolonged to the celestial sphere, will evidently meet the sphere 
in a circle if the star is at the pole of the ecliptic, and in an 
ellipse if it is not at the pole. 

In order to separate the effects of parallax and aberration, 
the following method was adopted by the astronomer Bessel. 
Instead of attempting to determine by direct observation the 
change of position of the star whose parallax was sought, he 
selected another star of much less magnitude, and therefore sup- 
posed to be at a much greater distance, which lay very nearly in 
the same direction as the first star, and observed the changes in 
the distance between these two stars and in the direction of the 
line joining them, during the year. Fig. 76 will serve to ex- 
plain the general principle of this a 
method. Let S be the position of the 
star whose parallax is sought, and s 
the position of the smaller star, both 
being projected on the surface of the 
celestial sphere. By the motion of the 
earth in its orbit the star S will describe 
the parallactic ellipse ADBC, and the 
star s, the ellipse adbc. When S ap- 
pears to be at A, s will appear to be at 
a; when S is at D, s will be at d, &c. 
It is evident that Aa and Bb will lie 




222 MAGNITUDE OF THE STARS. 

in different directions, and that Dd will be greater than Cc: and, 
therefore, by observing the different directions of the line joining 
the two stars, and also its different values, during the year, we 
may obtain the difference of parallax of the two stars, and, ap- 
proximately, the parallax of S. 

277. Results. — This method was applied by Bessel to the star 
61 Cygni. For the sake of greater accuracy he made use of 
two very small stars, situated very near to that star, whose ab- 
solute parallaxes he assumed to be equal. The parallax which he 
obtained for this star was 0".35. Other observations make it 
0".56: and the mean of these two values, or 0".45, corresponds 
to a distance of about forty-two trillions of miles, a distance 
which light would require seven years to traverse. The parallaxes 
of eleven other stars have been obtained with more or less accu- 
racy. The star about whose parallax there is the least doubt is « 
Centauri, which is probably the nearest to us of all the stars. Its 
parallax is 0".92: corresponding to a distance of about twenty- 
one trillions of miles. A table of these stars is given at the end 
of this book. 

According to the Russian astronomer Peters, the mean paral- 
lax of the stars of the first magnitude is 0".21, corresponding to 
a distance which light would traverse in 15i years. Another 
Russian astronomer, Struve, concludes that the distance of the 
most remote stars which can be seen in Lord Rosse's great tele- 
scope is about 420 times the distance of the stars of the first 
magnitude: from which the marvellous inference is drawn that 
the distance of the most remote telescopic stars from the earth is 
only traversed by light in 6500 years. 

A knowledge of the distance of a star, and of its proper mo- 
tion, enables us to estimate the amount in miles of its transverse 
motion. Sirius, for instance, has a proper motion of l".2o a year, 
and its parallax is 0".23. Hence its annual transverse motion is 
equal in amount to the radius of the earth's orbit multiplied by 
*A 5 • which is a motion of about sixteen miles a second. This is 
only the projection upon the celestial sphere of its real motion, 
which may be much greater. 

278. Real Magnitudes of the Stars. — Hitherto, when we have 
determined the distance of a celestial body, we have been able to 



VARIABLE STARS. 223 

compute its real diameter by means of observations made upon 
its angular diameter. But this method fails when we attempt to 
make use of it in obtaining the magnitude of the stars, since 
they do not present any measurable disc. It is true that with 
the better class of telescopes some of the stars appear to have a 
sensible disc; but this disc is really what is called a spurious 
one. This is proved by the fact that when a star is occulted by 
the moon, the size and the shape of the apparent disc remain un- 
altered up to the time of occupation, and its disappearance is then 
instantaneous. In the case of a solar eclipse, however, or of the 
occultation of a planet, the disappearance of the disc is gradual. 
We can, however, obtain some idea of the probable magni- 
tude of a star by comparing the light which it emits with that 
which is emitted by the sun. This comparison is made by 
means of the light of the moon. The ratio of the light of the 
sun to that of the full moon, and the ratio of the light of the full 
moon to that of some of the stars, have been obtained by appro- 
priate experiments; and, from a comparison of the results of 
these experiments, it is inferred that if the sun were removed to 
a distance from the earth equal to that of the nearest fixed star, 
it would appear only as a star of the second magnitude. The 
probability, then, is that unless there is a marked difference in 
the intensity of the light which these different bodies emit, the 
sun is not so large as most, and perhaps all, of the stars of the 
first magnitude. There is, however, as might be expected from 
the delicacy of the observations, some discrepancy in the results 
of these various photometric experiments : the light of Sirius, for 
instance, is said by some observers to be one hundred, and by 
others to be four hundred, times as great as the light of our sun 
would be, were it removed to a distance from us equal to that 
of Sirius. 

VARIABLE AND TEMPORARY STARS. 

279. Variable Stars. — There are certain stars which exhibit 
periodic changes in their brightness, the periods being in some 
cases only a few days in length, and in other cases embracing 
many years. The star o Ceti, called also Mir a, is an example 
of this class of stars. When brightest, it is a star of the second 



224 TEMPORARY STARS. 

magnitude. It remains in this state for about two weeks, and 
then begins to diminish in brightness, becoming wholly invisible 
to the naked eye in about three months, and appearing in tele- 
scopes as a star of the ninth or the tenth magnitude. After about 
five months it again appears, and in three months again reaches 
its maximum of brightness. The period in which these changes 
occur is 331 i days: at least, that is its mean value: its extreme 
values being 25 days more and 25 days less than this. It is also 
noticed that the rate of its increase and decrease of brightness is 
not always the same, and that one maximum of brightness is 
not always equal to another. 

Algol, or ft Persei, is another remarkable variable star. It 
remains for about sixty-one hours as a star of the second magni- 
tude. At the end of that time it begins to decrease in bright- 
ness, and becomes a star of the fourth magnitude in less than 
four hours. After about twenty minutes its brightness begins 
to increase, and another period of less than four hours brings it 
up again to a star of the second magnitude. 

The whole number of variable stars is above one hundred. 

280. Several theories have been advanced in explanation of 
this periodicity of brightness in the variable stars. One theory 
is that the surfaces of these stars are not uniformly luminous, and 
that therefore, in rotating upon their axes, they may present at 
one time the lighter portions of their surfaces to the earth, and 
the darker portions at another. Such a variation of luminosity 
might be caused by the presence of spots on the surfaces of the 
stars, similar to the spots on the sun, but of much greater extent. 
A second theory is, that nebulous bodies may revolve as satellites 
about these stars, and may intercept their light by coming in be- 
tween them and the earth. The irregularities noticed in the pe- 
riods of these stars, however, seem to constitute an objection to this 
second theory, while they may be allowed by the first theory, since 
analogous fluctuations in the periods and the magnitudes of the 
sun's spots have been observed. Arago suggests that, if it is 
true, as has been asserted by some astronomers, that these stars 
when at their minimum are surrounded by a kind of fog, the 
diminution of light may be due to the interference of clouds. 

281. Temporary Stars. — There are stars which have appeared 



DOUBLE STARS. 225 

at times in different parts of the heavens, and have afterwards 
disappeared. Such stars are called temporary stars. In compar- 
ing recent catalogues of stars with the catalogues of ancient as- 
tronomers, it is found that some stars which were formerly visible 
are no longer to be seen, while others which are now visible to 
the naked eye are not mentioned in the ancient catalogues. 
Some of these cases may be due to errors of observation, but 
hardly all of them. Moreover, similar instances have occurred 
in modern times. For example, it is recorded by the Danish 
astronomer, Tycho Brahe, that in November, 1572, a brilliant 
star suddenly blazed forth near the constellation Cassiopeia, and 
remained in sight about 17 months. When at its brightest 
phase, it equalled Venus in splendor, and was visible in broad 
daylight. It disappeared in 1574, and has never since been 
seen. Similar temporary stars are recorded as having appeared 
in or near the same place, in 945 and 1264. It is therefore 
possible that this may be really a variable star, and that it may 
reappear in the latter part of the present century. 

In May, 1866, a star of the ninth magnitude, in the constel- 
lation Corona Borealis, suddenly increased in brightness, and 
then rapidly decreased. On the 12th of the month it was of the 
second magnitude ; on the 14th, of the third ; and the rate of 
decrease in its brightness w T as for some time about half a mag- 
nitude a day. Lockyer says that there is good reason to believe 
that this sudden increase of brilliancy was due to the ignition 
of hydrogen in the star's atmosphere. 

It is very probable that these temporary stars are really in 
no respect different from variable stars, except in the length of 
their periods. Sir John Herschel says, with reference to these 
stars, that it is worthy of notice that all of them which are on 
record have been situated in or near the borders of the Milky 
Way. 

DOUBLE AND BINARY STARS. 



Castor. Rigel. Polaris. t Cassiop. 12 Lyncis. e Lyras. 

282. Double Stars. — Many of the stars which appear single 

15 



226 BINARY STARS. 

to the naked eye, are found, when examined in telescopes, to 
consist of two stars, apparently very near to each other. These 
are called doable stars. Only four were known to exist until near 
the close of the last century, wdien Sir William Herschel dis- 
covered about 500. The whole number of double stars now 
known exceeds 6000. In some cases the two stars are nearly 
of the same magnitude, but more frequently one of them is a 
large star, and the other a small one. Castor is an instance of 
the former class, and Sirius, Vega, and the pole-star are instances 
of the latter class. Some stars are found to consist of three, four, 
five, or more stars, and are called triple, quadruple, &c. stars. 
The star e Lyrse, for instance, appears in ordinary telescopes to 
consist of two stars, but with telescopes of greater power each 
of these stars is resolved into two others. 

283. Binary Stars. — The question arises with reference to these 
combinations : are the stars which compose them really con- 
nected, as are the sun and the planets; or is their appearance 
merely an optical illusion, arising from the fact that the stars 
in any one combination happen to lie in the same direction 
from the earth, although they may at the same time be at an 
immense distance from each other? The chances, at all events, 
are very much against the latter supposition. The astronomer 
Struve has calculated that there is only about one chance in 9570 
that, if the stars of the first seven magnitudes were scattered at 
random in the heavens, any two of them w T ould fall within 4" 
of each other; and yet more than 100 such cases have been 
observed. He has further calculated that there is only about one 
chance in 200,000 that three stars would accidentally fall within 
30" of each other, so as to form a triple star; and at least four 
such cases are to be found. 

The chances, then, are that the stars in these various combi- 
nations are physically connected. But more than this : it was 
announced by Sir William Herschel in 1803, after twenty-five 
years of observation upon many of the double stars, that in 
each double star which he had examined, the two stars of which 
it was composed revolved about each other in regular orbits, 
and in fact constituted a sidereal system. Subsequent observa- 
tions by other astronomers have fully verified this conclusion, 



BINARY STARS. 



127 



and about 600 double stars have been found to consist of stars 
revolving about each other, or rather about their common centre 
of gravity, according to the Newtonian law of gravitation. 
Such double stars are called binary stars, to distinguish them 
from other double stars, the components of which have not as 
yet been found to be physically connected. There are also 
other double stars, the components of which, while as yet they 
do not seem to revolve about each other, have constantly the 
same proper motion : thus showing that they are in all probability 
moving as one system through space. Triple, quadruple, &c. 
stars, whose constituents are found to be physically connected, 
are called ternary, quaternary, &c. stars. 

The orbits of the binary stars are found to be ellipses of con- 
siderable eccentricity. The periods of their revolutions have 
also been approximately determined, and extend over a very 
wide range. There are only about eight stars whose periods are 
less than 100 years, and only about 150 whose periods are less 
than 1000 years. 

284. Alpha Centaur i. — The star a Centauri, a star of the first 
magnitude in the southern hemisphere, is found to consist of 
two components, one of the first magnitude and the other of 
the second. The relative positions of these two components 

have been carefully noted during the last ., 

40 years. In Fig. 77, A represents one of 
these components, and BB'B" the apparent 
path of the other about it. The major axis 
of the orbit is about 30", and the period 75 
years. Its eccentricity is 0.95. 

Since the distance of a Centauri from the 
earth is approximately known, we can ob- 
tain some idea of the dimensions of the 
orbit. If Em the figure represents the earth, 
we shall have, in the right-angled triangle 
COE, the angle E equal to 15", and the side 
EO equal to 21 trillions of miles. Hence, 

CO = OE X tan 15" == 1,500,000,000 miles : 
which is nearly equal to the distance of Uranus from the sun, 
or to 16 times the radius of the earth's orbit. 



1S26 




228 



COLOKED STARS. 



Again, knowing the radius of the orbit and the period, we 
can obtain an approximate value of the mass of a Centauri 
from the formula in Art. 214. It will be found to be about 
T 7 oths of the mass of the sun. 

COLORED STARS. 

285. Many of the stars, both isolated and double, shine with 
a colored light. The isolated colored stars are usually red or 
orange : blue or green stars being very uncommon. The num- 
ber of red stars now recognized is at least 300, about 40 of 
which are visible to the naked eye. Among the most con- 
spicious of the red stars are Aldebaran and An tares ; and it is 
worthy of notice that nearly all the variable stars are of this 
color. According to Mr. Ennis's observations, Capella, Rigel, 
Procyon, and Spica are blue; Sirius, Vega, and Altair are green ; 
and Arcturus is yellow. 

The components of the double stars are often of different co- 
lors; blue and yellow, or green and yellow; and, less frequently, 
white and purple, or white and red. The following table con- 
tains a few of the many instances of such stars which might be 
given : — 



Name. 


Magnitude 

of 
Components. 


Color of the Larger. 


Color of the Smaller. 


V Cassiopeise 

a Piscium 

i Cancri 

s Bootis 


4 7 

5 6 

5 8 

3 7 

5 6 

3 7 


Yellow. 
Pale Green. 
Orange. 
Pale Orange. 
White. 
Yellow. 


Purple. 

Blue. 

Blue. 

Sea Green. 

Purple. 

Blue. 

-i 


C Coronse 


Cvsrni 





When the colors of the components are complementary, and the 
components are of very unequal size, it is possible that only one 
of the colors may really exist; the other being, according to a 
law of Optics, merely the result of contrast. Such an opinion is 
held by some astronomers ; but the objection is raised to it by 
others that, if one of these colors is only accidental, it ought to 
disappear when the eye is shielded from the light of the star 



PLATE IV. 




NEBULiE. 

1. ANNULAR NEBULA, 57 M LYR£. 

2. PLANETARY NEBULA, 3614 H VIRGINIS. 

3. NEBULOUS STAR, i ORIONIS. 

4. SPIRAL NEBULA, 99 M VIRGINIS. 

5. CRAB NEBULA IN TAURUS. 



NEBULiE. 229 

which has the other color: which, however, is very far from 
being the case. Another objection is that a similar phenomenon 
ought to be seen in all colored stars whose components are of 
unequal sizes : whereas many double stars are found in which 
both components have the same color, sometimes red and some- 
times blue. In one of Struve's catalogues, out of 596 double 
stars, there were : — 

295 pairs, both white: 
118 pairs, both yellowish or reddish: 
63 pairs, both bluish : 
120 pairs of totally different colors. 
In a few instances stars have changed their color. Sirius, 
for instance, is now green ; but both Ptolemy and Seneca ex- 
pressly state that in their day it had a reddish hue. Capella, 
which is now blue, was formerly red. Some observers state that 
seventeen stars of the first magnitude are colored, and that seven 
of these have changed their color. This change of color is par- 
ticularly noticeable in the variable stars. In that of 1572 (Art. 
281), the color changed from white to yellow, and then to a de- 
cided red ; and it is observed generally in the variable stars that 
the redness of the light increases as its intensity diminishes. 



CLUSTERS AND NEBUL.E. 

286. If we examine the heavens on any clear night when the 
moon is below the horizon, we shall find here and there groups 
of stars, which present a hazy, cloud-like appearance. These 
groups are classified into clusters and nebulae, although it is diffi- 
cult to establish any precise distinction between the two classes. 
When the different members of a group, or at all events some 
of them, can be separated from each other with the naked eye, the 
group is called a cluster. A nebula, on the other hand, is either 
wholly invisible to the naked eye, or, if seen at all, presents, as 
the name implies, only an ill-defined, cloudy appearance. The 
number of nebulae which have been discovered is over 5000. 
Some of these nebulse are resolved, with the aid of the telescope, 
into separate stars: while others, even when examined in the 
most powerful telescopes, preserve their cloudy appearance, and 



230 NEBULA. 

give no trace of stars. The former are called resolvable nebula?, 
the latter irresolvable nebulae. 

As every increase of telescopic power has rendered resolvable 
some of the nebulae previously classified as irresolvable, the com- 
mon opinion of astronomers, until within the last few years, was 
that there was no distinction between these nebulas more funda- 
mental than that of distance, magnitude, or intensity of light. 
Recently, however, the light of some of these nebulas has been 
subjected to spectroscopic analysis, and the results seem to show 
the existence of decidedly different classes of nebulas. The re- 
searches of Mr. Huggins show that the light of some of these 
nebulas gives spectra, which are distinguished, not by dark lines, 
but by bright ones : which shows, as we have already noticed 
(Art. 102), that the light does not come from solid bodies in a 
state of ignition, but from incandescent vapor or gas. Some at 
least, then, of the nebulas are to be considered as bodies of a 
gaseous nature ; and there are indications that the gases of which 
they are composed are hydrogen and nitrogen. The Great Ne- 
bula in Orion is an instance of this class of nebulas. 

The nebulas are very unequally distributed over the heavens. 
They congregate especially in a zone crossing the Milky AVay at 
right angles, and they are especially abundant in the constella- 
tions Leo, Ursa Major, and Virgo. 

287. Examples of Clusters. — The most noticeable cluster is the 
well-known group called the Pleiades. This group contains six 
or seven stars which are visible to the naked eye, the brightest 
star being of the third magnitude: and glimpses of many more 
can be obtained by examining the group with the eye turned 
sideways. With the aid of a telescope between one hundred and 
tw T o hundred stars can be detected. 

The Hyades are a group in Taurus, near the star Aldebaran, in 
which from five to seven stars can be counted. This group was 
supposed by the ancients to have some influence upon the rain. 

Prassepe, or the Bee-hive, in Cancer, is visible to the naked eye 
as a luminous spot. With a telescope of moderate power, more 
than forty conspicuous stars are seen within a space about 30' square, 
together with many smaller ones. Before the invention of the te- 
lescope, this group was probably one of the few recognized nebulas. 



NEBULA. 231 

288. Different Forms of Nebulce. — The groups which usually 
go by the name of nebulae present, as might be expected, almost 
every variety of form ; and most of these forms are too irregular 
to admit of any classification or description. Such is not the case, 
however, with all of them: and some of the different varieties of 
shape are exemplified in the following list: — 

1. Annular Nebulae ; 

2. Elliptic Nebulae; 

3. Spiral Nebulae; 

4. Planetary Nebulae ; 

5. Nebulous stars. 

There are also individual nebulae whose names indicate the 
general appearance which they present : such are the " Horse-shoe 
Nebula," the "Crab Nebula," the " Dumb-bell Nebula," &c. 

289. Annular Nebulce. — Of annular or ring-shaped nebulae, 
the heavens present four examples. The most remarkable one is 
situated in the northern constellation Lyra. Sir John Herschel 
says of it: "It is small, and particularly well-defined, so as 
to have more the appearance of a flat, oval, solid ring than of a 
nebula. The axes of the ellipse are to each other in the pro- 
portion of about four to five, and the opening occupies rather 
more than half the diameter. The central vacuity is filled in 
with faint nebulae, like a gauze stretched over a hoop. The 
powerful telescopes of Lord Rosse resolve this object into ex- 
cessively minute stars, and show filaments of stars adhering to 
its edges." 

290. Elliptic Nebulce. — There are several instances of elliptic 
or oval nebulae, of various degrees of eccentricity. The very 
conspicuous nebula called "the Great Nebula in Andromeda" is 
an example of this class. This object is distinctly visible to the 
naked eye, and is sometimes mistaken for a comet. When viewed 
with a telescope of moderate power, it has an elongated oval 
form ; but in the largest telescopes its aspect is very different 
from this. Professor G. P. Bond traced it through a length of 
4°, and a breadth of 2J°, and also discovered in it two curious 
black streaks, lying nearly parallel to the major axis of the 
oval. He also succeeded in detecting in it evidence of a stellar 
constitution. 



232 NEBULAE. 

Some elliptic nebulse are remarkable for having double stars 
at or near each of their foci. 

291. Spiral Nebulce — Observations with the Earl of Posse's 
telescope have led to the discovery of nebulse which consist 
of spiral bands proceeding from a common nucleus, and some- 
times from two nuclei. The most remarkable of these spiral 
nebulae is situated near the extremity of the tail of the Great 
Bear. It consists of nebulous coils diverging from two luminous 
centres, about 5' apart, and gives evidence of stellar composition. 

292. Planetary Nebulce. — Planetary nebulse received their name 
from Sir William Herschel, on account of the resemblance in form 
which they bear to the larger planets of our system. The out- 
lines of some of them are well defined, while those of others are 
indistinct; and some of them shine with a blue light. About 
twenty-five have been discovered, most of them being situated 
in the southern hemisphere. One of these nebulse is situated 
near ft Ursse Majoris, and has a diameter of 2' 40". It was dis- 
covered by Mechain, in 1781, and is described as "a very singular 
object, circular and uniform, which after a long inspection looks 
like a condensed mass of attenuated light." Perforations and a 
spiral tendency have been detected in it by the Earl of Posse. 

These planetary nebulae can hardly be globular clusters of 
stars, as they would in that case be brighter in the middle than 
at the borders, instead of being, as they are, uniformly bright 
throughout. Some astronomers suppose that they are assem- 
blages of stars, arranged either in cylindrical beds, or in the 
form of hollow spherical shells. 

293. Nebulous Stars. — Nebulous stars are stars which are sur- 
rounded by a faint nebulous envelope, usually circular in form, 
and sometimes several minutes in diameter. In some cases this 
envelope terminates in a distinct outline, while in others it gra- 
dually fades away into darkness. According to Hind, nebulous 
stars "have nothing in their appearance to distinguish them from 
others entirely destitute of such appendages; nor does the nebu- 
lous matter in which they are situated offer the slightest indica- 
tions of resolvability into stars, with any telescopes hitherto 
constructed." 

294. Double Nebulce. — M. D'Arrest, of Copenhagen, mentions 



VARIATION Or BRIGHTNESS. 233 

fifty double nebula? whose components are not more than 5' apart, 
and estimates that there may be as many as 200 such double 
nebula?. The two components are generally of a globular form. 
It is possible that these components may in some cases be 
physically connected. In one instance there seem to have been 
changes in the distance and the relative position of the two mem- 
bers during the last eighty years, which indicate the possibility 
of a revolution of one of the members about the other. 

295. Magellanic Clouds. — These two nebula?, called also nube- 
cula major and nubecula minor, are situated near the southern 
pole of the heavens. They are visible to the naked eye at night, 
when the moon is not shining, and have an oval shape. One 
of them covers an area of forty-two square degrees, the other an 
area of ten square degrees. Sir John Herschel found them to 
consist of swarms of stars, clusters, and nebula? of every 
description. 

296. Variation of Brightness in the Nebuke. — In the case of a 
few of the nebula? a change of brightness has been discovered, 
which is to some extent analogous to what has already been 
noticed in connection with the variable and the temporary stars. 
In 1852, Mr. Hind discovered a small nebula in the constella- 
tion Taurus. This nebula was observed from that time until 
the year 1856; but in 1861 it had entirely disappeared. In 
1862 it was seen in the telescope at Pulkowa, and for a short 
time appeared to be increasing in brightness; but towards the 
end of 1863, Mr. Hind, upon searching for it with the telescope 
with which it was originally discovered, was unable to find any 
trace of it. It is a curious coincidence that a star situated very 
near to this nebula has changed, in the same interval of time, 
from the tenth magnitude to the twelfth. 

There are four or five instances on record of similar changes 
in the brightness of nebulse. The nebula known as 80 of 
Messier's Catalogue, is said to have changed, in the months 
of May and June, 1860, from a nebula to a well-defined star 
of the seventh magnitude, and then to have resumed its ori- 
ginal appearance. The change was, however, so rapid and so 
decided, that some astronomers are inclined to ascribe it to the 
existence of a variable star between the nebula and the earth, 



234 MILKY WAY. 

rather than to a variation in the nebula itself. The Great Ne- 
bula in the constellation Argo, when observed by Sir John Her- 
schel in 1838, contained in its densest part the star rj, which was 
then of the first magnitude. Observations made in 1863, how- 
ever, showed that this star had become entirely free from its 
nebulous envelope, and had also been reduced to a star of the 
'sixth magnitude. The outline of certain parts of the nebula 
was also found to be different from what it had been represented 
to be by Herschel. More recent observations show still further 
changes in this nebula. 

297. The Galaxy, or Milky Way.— The Milky Way, that well- 
known luminous band which stretches through the heavens, may 
be considered to be a continuous resolvable nebula. If we follow 
the line of its greatest brightness, and neglect occasional devia- 
tions, we find its course to be nearly that of a great circle of the 
heavens, inclined to the equinoctial at an angle of about 63°. 
At one point a kind of branch is sent off, which unites again 
with the main stream at a distance of about 150° from the point 
of separation. The angular breadth of the belt varies from 6° 
to 20°, the average breadth being about 10°. When examined 
with the telescope, this band is found to be made up of thousands 
and millions of telescopic stars, grouped together with every 
degree of irregularity. Of the 20,000,000 stars of the first four- 
teen magnitudes, about nine-tenths are in or near the Milky Way. 
At some points the stars are so close to each other that they 
form one bright mass of light; while at other points there are 
dark spaces containing scarcely a star which is visible to the 
naked eye. A very noticeable break of this kind occurs in that 
part of the Milky Way which lies nearest to the south pole. 
This dark space is about 8° in length and 5° in breadth. It 
contains only one star visible to the naked eye, and that is a very 
small one. Early southern navigators gave to this vacancy the 
name of the Coal Sack. 

298. Number of the Stars. — Scarcely more can be said of the 
probable number of the stars, than that it is as inconceivably 
great as are the distances by which they are separated from the 
earth and from each other. Sir William Herschel states that 
on one occasion, while observing the stars in the Milky Way, he 



MOTION OF THE SOLAIi SYSTEM. 235 

estimated that 116,000 stars passed through the field of his tele- 
scope in fifteen minutes; and that, on another occasion, nearly 
260,000 stars passed in forty-one minutes. It may assist us in 
correctly appreciating the magnitude of these numbers to re- 
member that the number of stars visible to the naked eye in 
the whole heavens is less than 7000 (Art. 266). Results still 
more astonishing follow from an examination of the nebuke. 
Take, for instance, the "Great Nebula in Andromeda," which, 
by observations made with the telescope at Cambridge, Mas- 
sachusetts, within the last few years, has been found to give 
evidence of stellar composition. According to Professor G. P. 
Bond, its length is 4°, and its breadth 2 p. Now, if a telescope 
just suffices to resolve a nebula into separate stars, it is safe to 
assume that the angular distance between any two contiguous 
stars in it cannot be greater than 1". If, then, we multiply to- 
gether the number of seconds in the length of this nebula, and 
the number of seconds in its breadth, we obtain over 129,000,000 
for an approximate estimate of the number of stars in this one 
nebula. 

MOTION OF THE SOLAR SYSTEM FN SPACE. 

299. We have now accustomed ourselves to consider the earth 
to be a planet, rotating upon a fixed axis once in every sidereal 
day, and revolving about the sun once in every sidereal year; but 
we have up to this point considered the sun and the solar sys- 
tem to be at rest in space. There are, however, observations 
which tend to show that such is not, in truth, the case, but 
that the whole solar system is in rapid motion through space. 
Analogy certainly supports such a conclusion. We have already 
seen that the solar system, immense as it may seem to be in 
itself, must be considered as scarcely more than a point in com- 
parison with the entire celestial system (Art. 275); and we 
have also seen reasons for concluding (Art. 278) that there are 
many bodies among the stars which are very much larger 
than the sun. There is, then, no reason for supposing that the 
solar system is the centre of the celestial system; aud there is 
nothing violent in the conclusion that, just as Jupiter revolves 
about the sun, and carries its satellites with it, so the sun may 



236 



MOTION OF THE SOLAR SYSTEM. 



revolve about some other body or point, and carry its system of 
planets and satellites with it. 

300. The Apparent Motions of the Stars. — The proper motions 
of the stars, already mentioned, may be explained in three dis- 
tinct ways. First, they may be what they seem to be, real 
motions of the stars, performed in orbits of immense radii; or, 
secondly, they may be only apparent motions, caused by a change 
of the position of the solar system in space; or, thirdly, they 
may be the results of both these causes existing together. The 
last of these theories was advanced by Sir William Herschel, 
in 1783; and although doubts were afterwards expressed as to 
its truth, later observations have tended to support it, and it is 
now generally accepted by astronomers. If we suppose for a 
moment that the solar system is approaching any given point 
in the heavens, the pole-star, for instance, the result will be that 
the stars which lie about that point will appear to separate 
slowly from it, while the stars which lie about the diametrically 




opposite point will appear to close up around that point; and in- 
deed all the stars will apparently move in an opposite direction 
to that in which we suppose the solar system to be moving, those 



MOTION OF THE SOLAR SYSTEM. 237 

stars having the greatest motion which lie in a direction at right 
angles to the direction of the motion of the solar system. In 
Fig. 78, let E be the position in space of the earth at any time, 
and let P,A,B,D, and G be stars supposed to be situated at equal 
distances from the earth. Let the earth, by the motion of the 
whole solar system in space, be carried to some point E', in the 
direction of the star P. It is evident that the angular distance 
of the star A from P, when the earth is at P", or the angle PEA, 
is greater than the angular distance between A and P when the 
earth is at E, or the angle PEA. In other words, while the 
earth has moved from E to E\ the star A has apparently receded 
from P. So also the star D has apparently approached the star 
0. The stars B and G have both receded from P, the retro- 
gradation of (?, or the angle EGE\ being greater than the 
retrogradation of P, or the angle EBE' '. 

301. Direction and Amount of this Motion. — The elements, 
then, of the motion of the solar system through space are deter- 
mined from what are called the proper motions of the stars. 
Recent observations and calculations have not only confirmed 
Herschel's theory that such a motion really exists, but have also 
very nearly confirmed his conclusion as to the point towards which 
the motion is directed. Different astronomers give different 
positions to this point ; but they all agree in the general conclu- 
sion, that the solar system is at present moving in the direction 
of a point in the constellation Hercules, situated in about 171 
hours of right ascension, and 35° of north declination. 

The calculations of M. Otto Struve lead him to the conclu- 
sion that for a star situated at right angles to the direction of 
the motion of the solar system and at the mean distance of the 
stars of the first magnitude, the annual angular displacement 
of the star due to that motion is 0."34: that is to say, the clis 
tance through which the system moves in one year subtends at 
the star an angle of that amount. ISTow, the mean parallax of 
the stars of the first magnitude, or the angle subtended at the 
mean distance of those stars by the radius of the earth's orbit, is 
estimated to be 0."21 (Art. 277); hence the annual motion of the 
solar system is the radius of the earth's orbit multiplied by ff, 
or about 150,000,000 miles: a little less than Hve miles a second. 



23$ ORBIT OF THE SOLAR SYSTEM. 

This is the estimate commonly accepted by astronomers. Mr. 
Airy, however, deduces a motion of about twenty-seven miles a 
second. 

302. The Orbit of the Solar System. — Although the motion of 
the solar system through space appears to be rectilinear, it does 
not follow that such is actually the case: since it may move in 
an elliptical orbit, with a radius vector so immense that years 
and even centuries of observation will be needed to show any 
sensible change of direction. The probability is that the sun, 
with its planets and their satellites, revolves about the common 
centre of gravity of the group of stars of which it is a member; 
and that this centre of gravity is situated in or near the plane 
of the Milky Way. If we suppose the orbit in which our 
system is moving to be an ellipse with a small eccentricity, then 
the centre of this ellipse will lie in a direction which makes an 
angle of about 90° with the direction in which the system is now 
moving. Madler concluded that Alcyone, the brightest star in 
the Pleiades, was the central sun of the celestial sphere, while 
Argelander, believing that this central point must lie in the 
principal plane of the Milky Way, places it in the constellation 
Perseus. It may be noticed, in connection with this subject, that 
Sir William Herschel was inclined to believe that some of the 
more conspicuous of the stars, such as Sirius, Arcturus, Capella, 
Vega, and others, were in a great degree out of the reach of 
the attractive power of the other stars, and were probably, like 
the sun, centres of systems. 

303. Years, and, in all probability, ages of observation will 
be needed to determine the elements of this most stupendous 
orbit. Madler's speculations have led him to the conclusion 
that the period of this revolution is 29,000,000 years : a period 
in comparison with which the years through which astronomical 
observations have extended are utterly insignificant. The stu- 
dent who is curious to know more of this subject will find the 
details fully set forth in Grant's History of Physical Astronomy. 
Grant himself says, in reference to the subject: — "It is manifest 
that all such speculations are far in advance of practical astro- 
nomy, and therefore they must be regarded as premature, how- 
ever probable the supposition on which they are based, or how- 



REAL MOTIONS OF THE STARS. 239 

ever skilfully they may be connected with the actual observation 
of astronomers." 

DETERMINATION OF THE REAL MOTIONS OF THE STARS. 

304. We have already seen (Art. 265), that in order to deter- 
mine the real motion of a star in space, we must be able to 
determine, not only its transverse motion, which is indicated by 
a change in its apparent position upon the celestial sphere, but 
also its motion directly to or directly from the earth. Now, a 
star situated at the nearest distance of the fixed stars, and 
moving towards the earth with a velocity equal to that of the 
earth in its orbit (18 miles a second), would diminish its dis- 
tance from us by only about ^th in a thousand years. The 
detection of any such motion by a change in the star's apparent 
brightness is, therefore, utterly out of the question. The spectro- 
scope, however, gives us quite another means of detecting such 
a motion. 

305. Analogy Between Light and Sound. — A clear conception 
of the principle upon which this method rests may be more 
readily obtained if we first notice the analogy between the con- 
duction of light and that of sound. Sound is the result of a 
series of waves or pulses in the air, produced by the vibrations 
of a sonorous body ; light is the result of a similar series of 
pulses or waves in the luminiferous ether, caused by the vibra- 
tions of the particles of a luminous body. The more rapidly 
a sonorous body vibrates, the greater will be the number of 
pulses or waves which it communicates to the air in the unit of 
time, and, consequently, the higher will be the pitch of the tone 
produced. In the case of a luminous body, the greater the 
number of waves in the luminiferous ether which the vibrations 
of any particle cause in the unit of time, the greater will be the 
refrangibility of the ray produced ; in the solar spectrum, for 
instance, the violet rays are the most refrangible of all the rays 
which we can see, and the number of waves in the unit of time 
necessary to produce them is greater than the number necessary 
to produce rays of any other color. The refrangibility of a ray 
is therefore analogous to the pitch of a tone. 

306. Change of Tone or Refrangibility. — It is proved by ex- 



240 REAL MOTIONS OF THE STARS. 

periment that if the distance between a sonorous body, producing 
a tone of constant pilch, and the hearer, is diminished by the 
motion of either, with a velocity that has an appreciable ratio to 
that of sound, the pitch of the tone will appear to be heightened ; 
and that if, on the contrary, the distance between the two is in- 
creased with sufficient rapidity, the pitch of the tone will appear 
to be lowered. So, too, in the case of light : if the luminous 
body and the observer approach each other, the refrangibility 
of the rays of light will be increased ; and if they separate, it 
will be diminished. If, then, we can detect any change of re- 
frangibility in the light of a heavenly body, we may conclude 
that the distance of that body from the earth is changing. 

307. Change of Refrangibility Detected. — Father Secchi, in a 
communication to the Comptes Rendus in the early part of 1868, 
after stating the principles of this method, announced that he 
had subjected the light of Sirius and of other prominent stars to 
an examination with the spectroscope, but had detected no evi- 
dence of motion. Since then, Mr. Huggins, an English observer, 
who has devoted himself especially to spectroscopic investiga- 
tions, has succeeded in detecting such a motion in Sirius. There 
is a certain dark line F in the blue of the solar spectrum, which 
corresponds to a bright line in the spectrum of hydrogen ; and 
this same line also appears in the spectrum of Sirius. Now, 
if Sirius has no motion either towards or from the earth, the 
line F in its spectrum will coincide in position with the corres- 
ponding line in the spectrum of hydrogen, when the two spectra 
are compared by means of the spectroscope: while if, on the 
contrary, Sirius has such a motion, its whole spectrum, lines 
and all, will be shifted bodily, and the line F will no longer 
coincide with the corresponding line in the spectrum of hydrogen. 

Mr. Huggins, using a spectroscope of large dispersive power, 
and carefully comparing the spectrum of Sirius with that of 
hydrogen, finds that the line F in the spectrum of Sirius is 
displaced, by about ^|o^ n °f an mcn - This displacement is 
towards the red end of the spectrum, and indicates that the 
refrangibility of the light of Sirius is diminished : since the 
red rays are the least refrangible of all the colored rays of the 
spectrum. Sirius is therefore receding from the earth. 



REAL MOTIONS OF THE STARS. 241 

308. Amount of the Heal Motion of Sirius. — The amount of 
recession corresponding to a displacement of this extent, when 
observed in a spectroscope whose power is equal to that of the 
one used by Mr. Huggins, is computed to be about 41 i miles 
a second. But it happens that when the observation was 
made, the earth was so situated in its orbit that it was receding 
from Sirius, by its revolution in its orbit, at the rate of about 
12 miles a second. The motion of the solar system in space, 
computed to be five miles a second (Art. 301), must also be 
taken into consideration ; and the point towards which this 
motion is at present directed is almost exactly opposite to the 
position of Sirius on the celestial sphere. The earth was there- 
fore moving away from Sirius at the rate of about 17 miles a 
second. If, then, we diminish the whole amount of the increase 
of distance between Sirius and the earth in one second, by the 
amount of the motion of the earth through space in one second, 
or 17 miles, we find that Sirius is moving through space, away 
from the earth, at the rate of about 24^ miles a second. 

By combining this motion with the transverse motion of 
Sirius, we can obtain an approximate value of its real motion. 
In Art. 277, the transverse motion of Sirius was computed to 
be about 16 miles a second. The resultant of these two motions 
is a motion of about 29 miles a second : or, 900,000,000 miles 
a year. 

Mr. Huggins has also made similar observations upon the 
Great Nebula in Orion, and other gaseous nebulae, without, how- 
ever, being able thus far to detect any motion in them. 

309. The numerical results of the preceding article may not 
be beyond criticism ; but the grand fact remains, that in all 
probability the motions of these distant bodies, which have so 
long seemed to be wrapped in hopeless mystery, are soon to 
come within the reach of our observation. The knowledge of 
these motions will be a powerful auxiliary in the determination 
of the law which undoubtedly binds all the heavenly bodies 
together in one great system ; and it is not presumptuous to 
expect that at some future day — no one can say how distant or 
how near — this law will be revealed to us. 

16 



242 



APPENDIX. 



APPENDIX. 



MATHEMATICAL DEFINITIONS AND FORMULA. 



PLANE TRIGONOMETRY. 



1. The complement of an angle or arc is the remainder ob- 
tained by subtracting the angle or arc from 90°. 

2. The supplement of an angle or arc is the remainder ob- 
tained by subtracting the angle or arc from 180°. 

3. The reciprocal of a quantity is the quotient arising from 

dividing 1 by that quantity : thus the reciprocal of a is 

4. In the series of right triangles ABC, AB'C\ AB" C" , &c, 
e"i — -,b" Fig. 79, having a common angle A, we have by 

Geometry, 

BC B'C B"C" 




AB~ 


~ AB' 


~ AB" 


BC 


B'C 


B"C" 


AC 


~ AC 


~1KC T 


AB 


AB' 


AB" 


AC 


~ AC 


~~ AC"' 



Fig. 79, 



The ratios of the sides to each other are there- 
fore the same in all right triangles having the 
same acute angle : so that, if these ratios are known in any one 
of these triangles, they will be known in all of them. These 
ratios, being thus dependent only on the value of the angle, 
without any regard to the absolute lengths of the sides, have 
received special names, as follows : 

The sine of the angle is the quotient of the opposite side 

BC 



divided by the hypothenuse. Thus, sin A 



AB' 



The tangent of the angle is the quotient of the opposite side 

BC 



divided by the adjacent side. Thus, tan A = 



AC 



TRIGONOMETRY. 243 

The secant of the angle is the quotient of the hypothenuse 

AB 

divided by the adjacent side. Thus, sec A = . - 

~A- C 

5. The cosine, cotangent, and cosecant of the angle are respec- 
tively the sine, tangent, and secant of the complement of the 
angle. Now, in Fig. 79, the angle ABC is evidently the com- 
plement of the angle BAC. Hence we have, 

cos A = sin B — - -—f) ; 

AC 
cot A = tan B = ^rr,: 

eosec A = see B = -~^ ■ 

6. Since the reciprocal of j- is — , we see, from the preceding 

definitions, that the sine and the cosecant of the same angle, the 
tangent and the cotangent, the cosine and the secant, are re- 
ciprocals. 

7. If two parts of a plane right triangle in addition to the right 
angle are given, one of them being a side, the triangle can be 
solved: that is to say, the values of the remaining parts can be 
obtained. This solution is effected by means of the definitions 
above given. 

8. When an angle is very small, its sine and its tangent are 
both very nearly equal to the arc which subtends the angle in 
the circle whose radius is unity. Hence, to find the sine or the 
tangent of a very small angle approximately, we have, if x is a 
small angle expressed in seconds, 

sin x = tan x = x sin 1", 
If x is expressed in minutes, 

sin x = tan x = x sin 1'. 

9. If x and y are any two small angles, we have from the pre- 
ceding article, 

sin x x sin 1" x 

sin y y sin 1" y ' 
that is, the sines (or tangents) of very small angles are propor- 
tional to the angles themselves. 

10. Cos x — 1 — 2 siri* i x. 



244 



APPENDIX. 



11. The sides of a plane triangle are proportional to the sinea 
of their opposite angles. 

12. In order to solve a plane oblique triangle, three parts must 
be given, and one of them must be a side. 

SPHERICAL TRIGONOMETRY. 

13. A spherical triangle is a triangle on the surface of a sphere, 
formed by the arcs of three great circles of the sphere. 

14. In a spherical right triangle, the sine of either oblique 
angle is equal to the quotient of the sine of the opposite side, 
divided by the sine of the hypothenuse. Thus, in the triangle 

B ABC, right-angled at C, Fig. 80, we have, 

sin B C 




sin 



A = 



Fig. 80. 



hypothenuse. 



sin AB 

15. The cosine of either oblique angle is 
equal to the quotient of the tangent of the 
adjacent side, divided by the tangent of the 
Thus, 

. tan A C 

eos A = — • 

tan AB 

16. The tangent of either oblique angle is equal to the quo- 
tient of the tangent of the opposite side, divided by the sine of 
the adjacent side. Thus, 

tan BC 

tan A = j-p, • 

sin AC 

17. A spherical right triangle can be solved when any two 
parts in addition to the right angle are given. The solution is 
effected by means of the formulae given in the preceding articles. 

ANALYTIC GEOMETRY. 

18. The circle, the ellipse, the hyperbola, and the parabola are 
often called conic sections; since each 
curve may be formed by intersecting 
a right cone by a plane. 

19. An ellipse is a plane curve, in 
which the sum of the distances of each 
point from two fixed points is equal to 
a given line. Thus, in Fig. 81, the 




ANALYTIC GEOMETRY. 



245 



sum of the distances of the point M from the two fixed points F 
and F' is always constant, wherever on the curve A CBD the 
point M may be situated. 

The two fixed points are called the foci of the ellipse, and the 
middle point of the line which joins them is called the centre. 

A line drawn from either focus to any point of the curve is 
called a radius vector. 

A line drawn through the centre, and terminating at each ex- 
tremity in the curve, is called a diameter. That diameter which 
passes through the foci is called the transverse or major axis: 
and that diameter drawn perpendicular to the transverse axis is 
called the conjugate or minor axis. Thus, AB is the transverse 
axis, and CD the conjugate axis. The transverse axis is equal 
to the constant sum of the distances FM and F'M. 

The eccentricity of the ellipse is the quotient of the distance 
from the centre to either focus, divided by half the major axis. 

OF . 

Thus, yrjL is the eccentricity. 

20. The solid generated by the revolution of an ellipse about 
its major axis is called a prolate spheroid: that generated by its 
revolution about its minor axis is called an oblate spheroid. 

The expression for the volume of an oblate spheroid is |-a 2 6: 
in which a and b denote the semi-major and the semi-minor axis 
of the generating ellipse, and k the ratio of the circumference 
of a circle to its diameter, or 3.1416. 

The compression, or oblateness, of an oblate spheroid is the 
ratio of the difference between the major and the minor axis of 
the generating ellipse to the major axis. 

21. The hyperbola is a plane curve, in which the difference of 
the distances of each point 
from two fixed points is 
equal to a given line. 
Thus, in Fig. 82, the dif- 
ference of the distances of 
any point M of the curve 
from the two fixed points 
jF and F' is equal to a 
given line. 




246 



APPENDIX. 



The two fixed points are called the foci, and the middle point 
of the line joining them is called the centre. 
22. The parabola is a plane curve, every 
point of which is equally distant from a 
fixed point, and from a right line given in 
the plane. Thus, in Fig. 83, in which AB 
is the given line, and F the given point, the 
distances FM and GM are equal to each 
other, for any point M of the curve. 
Fig. 83. The fixed point is called the focus. 




MECHANICS. 

23. The resultant of tw r o or more forces is a force which singly 
will produce the same mechanical effect as the forces themselves 
jointly. 

The original forces are called components. In all statical in- 
vestigations the components may be replaced by their resultant, 
and vice versa. 

24. If two forces be represented in magnitude and direction 
by the two adjacent sides of a parallelogram, the diagonal will 
represent their resultant in magnitude and direction. Thus, in 

B r Fig. 84, if a force act at A 

in the direction AD', and a 
second force act at A in the 
direction AB\ these two forces 
being represented in magni- 
tude by the lengths of the 
lines AD and AB respectively, 
the resultant of these two 
forces will be a force in the 
direction A C, and of a magnitude represented by the length 
of the line A C. The parallelogram is called the parallelogram 
of forces. 

25. Conversely: any given force may be resolved into two 
component forces, acting in given directions. Thus, in Fig. 84, 
let A C be the given force, and A B r and AD' the given directions. 
From C draw CB parallel to AD', and CD parallel to AB'. 




kirkwood'b law. 247 

AB and AD will be the two components acting in the given 
directions. 

26. The force which must continually urge a material point 
towards the centre of a circle, in order that it may describe the 
circumference with a uniform velocity, is equal to the square of 
the linear velocity divided by the radius. This proposition is 
expressed in the following formula ; 

j r 
v being the space passed over in the unit of time, r the radius of 
the circle, and / the magnitude of the force. 

27. The force which constantly urges a body towards the 
centre of its circular path is called a centripetal force. The 
tendency which the body has to recede from the centre, in con- 
sequence of its inertia, or the resistance which it offers to a de- 
flection from a rectilinear path, the resistance being estimated 
in the direction of the radius, is called a centrifugal force. These 
two forces are in equilibrium as long as the body moves in the 
same circular path, and the same expression serves for the mea- 
sure of each force. 

28. Let t be the periodic time, or the time of one revolution. 
We shall evidently have, 

vt = 2~r; 



Substituting this value of v 2 in the formula in Art. 26 above, we 
shall have, 

, 4-V 



KIRKWOOD'S LAW. 



In 1848, Daniel Kirkwood, of Pennsylvania, announced the 
discovery of an analogy in the distances, masses, periods of ro- 
tation, and periods of revolution of the principal planets. This 
analogy is known under the name of Kirkwood 9 s Law. The 



248 APPENDIX. 

original statement of it will be found in the American Journal 
of Science, Vol. ix., Second Series, and is as follows: 

"Let P be the point of equal attraction between any planet 
and the one next interior, the tw T o being in conjunction: P' that 
between the same and the one next exterior. 

"Let also D = the sum of the distances of the points P, P' 
from the orbit of the planet, which I shall call the diameter 
of the sphere of the planet's attraction : 

" If = the diameter of any other planet's sphere of attraction 
found in like manner: 

"n — the number of sidereal rotations performed by the 
former during one revolution around the sun: 

u nf — the number performed by the latter; then it will be 
found that 



: V? 3 ; or n = n! ( — j 



That is, the square of the number of rotations made by a planet 

during one revolution round the sun y is proportional to+the cube 

n 
of the diameter of its sphere of attraction: or jy^ is a constant 

quantity for all the planets of the solar system." 

This analogy, when first announced, created much interest 
among scientific men, many of whom considered that, if its truth 
were established, it would be a powerful argument in favor of the 
nebular hypothesis. There is so much uncertainty attending the 
determination of the masses and the periods of rotation of many 
of the planets, that the statement of this analogy has been ex- 
eluded from the main text of this book. 



ASTRONOMICAL CHRONOLOGY. 

The science of Astronomy seems to have been cultivated in 
very early ages by almost all the Eastern nations, particularly 
by the Egyptians, the Persians, the Indians, and the Chinese. 
Records of observations made by the Chaldseans as far back as 
2234 B.C. are said to have been found in Babylon. The study 
of Astronomy was continued by the Chaldeans, and in later times 



ASTRONOMICAL CHRONOLOGY. 249 

by the Greeks and the Romans, until about 200 a.d. After that 
time it was neglected for about six centuries, and was then re- 
sumed by the Eastern Saracens after Bagdad was built. It was 
brought into Europe in the thirteenth century by the Moors of 
Barbary and Spain. A full account of the rise and the progress 
of the study of Astronomy, is given in Vince's Astronomy, 
Vol. II. 

The instrument principally used by the ancient astronomers 
seems to have been a sort of quadrant, mounted in a vertical 
position. Ptolemy describes one which he used in determining 
the obliquity of the ecliptic. The Arabian astronomers had one 
with a radius of 21 § feet, and a sextant with a radius of 57 f feet. 

The following dates, with scarcely any change or addition, 
have been taken from George F. Chambers's Descriptive Astro- 
nomy (Oxford, 1867), in which many other interesting astrono- 
mical dates, here omitted, may be found. 

B.C. 

720. Occurrence of a lunar eclipse, recorded by Ptolemy. It 
seems to have been total at Babylon, March 19, 9?h. p.m. 

640-550. Thales, of Miletus, one of the seven wise men of Greece. 
He declared that the earth was round, calculated solar 
eclipses, discovered the solstices and the equinoxes, and re- 
commended the division of the year into 365 days. 

585. Occurrence of a solar eclipse, said to have been predicted 
by Thales. 

580-497. Pythagoras, of Crotona, founder of the Italian Sect. 
He taught that the planets moved about the sun in elliptic 
orbits, and that the earth was round; and is said to have 
suspected the earth's motion. 

547. Anaximander died. He asserted that the earth moved, and 
that the moon shone by light reflected from the sun. He 
introduced the sun-dial into Greece. 

500. Parmenides, of Elis, taught the sphericity of the earth. 

490. Alcmseon, of Crotona, asserted that the planets moved from 
west to east. 

450. Diogenes, of Apollonia, stated that the inclination of the 
earth's orbit caused the change of seasons. Anaxagoras is 
said to have explained eclipses correctly. 



250 APPENDIX, 

B.C. 

432. Meton introduced the luni-solar period of 19 years. He 
observed a solstice at Athens in 424. 

384-322. Aristotle wrote on many physical subjects, including 
Astronomy. 

370. Eudoxus introduced into Greece the year of 365J days. 

330. Callippus introduced the luni-solar period of 76 years. 
Pytheas measured the latitude of Marseilles, and showed 
the connection between the moon and the tides. He also 
showed the connection of the inequality of the days and 
nights with the differences of climate. 

320-300. Autolychus, author of the earliest works on Astronomy 
extant in Greek. 

306. First sun-dial erected in Rome. 

287-212. Archimedes observed solstices, and attempted to mea- 
sure the sun's diameter. Aristarchus declared that the 
earth revolved about the sun, and rotated on its axis. 

276-196. Eratosthenes measured the obliquity of the ecliptic, 
and found it to be 20 J degrees. He also measured a degree 
of the meridian with considerable exactness. 

190-120. Hipparchus, called the Newton of Greece. He dis- 
covered precession : used right ascensions and declinations, 
and afterwards latitudes and longitudes: calculated eclipses: 
discovered parallax : determined the mean motions of the sun 
and the moon ; and formed the first regular catalogue of the 
stars. 

50. Julius Caesar, with the Egyptian astronomer Sosigenes, under- 
took to reform the calendar. 

A.D. 

100-170. Ptolemy, of Alexandria, author of the Ptolemaic Sys- 
tem of the Universe, in which the earth is the centre. He 
wrote descriptions of the heavens and the Milky Way, 
and formed a catalogue giving the positions of 1022 fixed 
stars. He appears to have been the first to notice the re- 
fraction of the atmosphere. 

642. The School of Astronomy, at Alexandria, founded ten 
centuries previously by Ptolemy the Second, was destroyed 
by the Saracens under Omar. 

762. Rise of Astronomy among the Eastern Saracens. 



ASTRONOMICAL CHRONOLOGY. 251 



A.D. 



880. Albatani, the most distinguished astronomer between Hip- 
parchus and Tycho Brahe. He corrected the values of pre- 
cession and the obliquity of the ecliptic, formed a catalogue 
of the stars, and first used sines, chords, &c. 

1000. Abul-Wefa first used secants, tangents, and cotangents. 

1080. The use of the cosine introduced by Geber, and also some 
improvements in Spherical Trigonometry. 

1252. Alphonso X., King of Castile, under whose direction cer- 
tain celebrated astronomical tables, called the Alphondne 
Tables, were compiled. 

1484. Waltherus used a clock with toothed wheels. (The earliest 
complete clock of which there is any certain record was 
made by a Saracen in the thirteenth century.) 

1543. Publication of Copernicus's De Revolutionibus Orbium Ce~ 
lestium, setting forth the Copernican System of the Universe. 

1581. Galileo determined the isochronism of the pendulum. 

1582. Tycho Brahe commenced astronomical observations near 
Copenhagen. 

1594. Gerard Mercator, author of Mercator's Projection. (The 

date is doubtful, and may have been as early as 1556.) 
1576. Fabricius discovered the variability of o Ceti. 

1603. Bayer's Maps of the Stars published. 

1604. Kepler obtained an approximate value of the correction 
for refraction. 

1608. Jansen and Lippersheim, of Holland, are said to have 
invented the refracting telescope, using a convex lens. It 
is, however, a disputed point as to who really invented the 
telescope. The use of the lens seems to have been known 
about fifty years before this. 

1609. Kepler announced his first two laws. 

1610. Galileo, using a telescope with a concave object-lens, dis- 
covered the satellites of Jupiter, the librations of the moon, 
the phases of Venus, and some of the nebulae. 

1611. Spots and rotation of the sun discovered by Fabricius. 
1614. Napier invented logarithms. 

1618. Kepler announced his third law. 

1631. The first recorded transit of Mercury, observed by Gas- 
sendi. The vernier invented, 



252 APPENDIX. 

A.D. 

1633. Galileo forced to deny the Copernican theory. 

1639. First recorded transit of Venus, observed by Horrox and 
Crabtree. 

1640. Gascoigne applied the micrometer to the telescope. 
1646. Fontana observed the belts of Jupiter. 

1654. The discovery of Saturn's rings by Huyghens. (Galileo 
had previously noticed something peculiar in the planet's 
appearance.) 

1658. Huyghens made the first pendulum-clock. (The discovery 
is also ascribed to Galileo the younger.) 

1662. Royal Society of London founded. 

1663. Gregory invented the Gregorian reflecting telescope. 

1664. Hook detected Jupiter's rotation. 

1666. Foundation of the Academy of Sciences at Paris. 
1669. Newton invented the Newtonian reflecting telescope. 

1673. Flamsteed explained the equation of time. 

1674. Spring pocket-watches invented by Huyghens. (Said also 
to have been invented, somewhat earlier, by Dr. Hooke.) 

1675. Transmission of light discovered by Homer. Transit In- 
strument used to determine right ascensions by Homer. 
Greenwich Observatory founded. 

1687. Newton's Principia published. 

1690. Ellipticity of the earth theoretically determined by Huy- 
ghens. 

1704. Meridian Circle used by Romer. 

1711. Foundation of the Royal Observatory at Berlin. 

1725. Compensation pendulum announced by Harrison. 

1726. Mercurial pendulum invented by Graham. 

1727. Aberration of light discovered by Bradley. 
1731. Hadley's Quadrant invented. 

1744. Euler's Theoria ' Motuum published, the first analytical 
work on the planetary motions. 

1745. Nutation of the earth's axis discovered by Bradley. 
1750. Wright's Theory of the Universe published. 

1765. Harrison rewarded by Parliament for the invention of the 

Chronometer. 
1767. British Nautical Almanac commenced. 
1769. Transit of Venus successfully observed. 



ASTRONOMICAL CHRONOLOGY. 253 

A.D. 

1774. Experiments by Maskelyne to determine the earth's den- 
sity. 

1781. Uranus discovered by Sir William Herschel. Messier's 
catalogue of Nebulae published. 

1783. Herschel suspected the motion of the whole solar system. 

1784. Researches of Laplace on the stability of the solar system. 

1786. Publication of Herschel's catalogue of 1000 nebula?. 

1787. Herschel began to observe \vith his forty-feet reflector. The 
Trigonometrical Survey of England commenced. 

1788. Publication of La Grange's Mecanique Analytique. 

1789. The rotation of Saturn determined by Herschel, and a 
second catalogue of 1000 nebulae published. 

1792. Commencement of the Trigonometrical Survey of France. 
1796. French Institute of Science founded. 
1799. Laplace's Mecanique Celeste commenced. 
1801-7. The minor planets Ceres, Pallas, Juno, and Vesta dis- 
covered. 

1802. Publication of Herschel's third catalogue of Nebulae. 

1803. Publication of Herschel's discovery of Binary Stars. 

1804. Proper motions of 300 stars published by Piazzi. 

1805. Commencement of attempts to determine the parallax of 
the stars. 

1812. Troughton's Mural Circle erected at Greenwich. 

1820. Foundation of the Royal Astronomical Society of London. 

1821. Observatory at Cape of Good Hope founded. 
1823. Cambridge (England) Observatory founded. 
1835. Airy determined the time of Jupiter's rotation. 

1837. Value of the moon's equatorial parallax determined by 
Henderson. The East Indian arc of 21° 21' completed. 

1838. Parallax of 61 Cygni determined by Bessel. 

1839. Parallax of a Centauri determined by Henderson. Im- 
perial Observatory at Pulkowa (Russia) founded. 

1840. Harvard Observatory founded. 

1842. Washington Observatory founded. Mean density of the 
earth determined by Baily. 

1843. Periodicity of the solar spots detected by Schwabe. 
18-44. Taylor's catalogue of 11,015 stars. Transmission of time 

by electric signals commenced in the United States. 



254 APPENDIX. 

A.D. 

1845. Discovery of the fifth minor planet, Astrsea. (101 others 
have since been discovered.) 

1846. Discovery of the planet Neptune. 

1847. Lalande's catalogue of 47,390 stars republished by the 
British Association. 

1851. Foucault's pendulum experiment to demonstrate the earth's 
rotation. Completion of the Russo-Scandinavian arc. 

1854. Difference of longitude of Greenwich and Paris deter- 
mined by electric signals. 

1855. Commencement of the American Ephemeris. 

1858. De La Rue obtained a stereoscopic view of the moon. 
(The first photograph of the moon was made by Dr. J. W. 
Draper, of New York, in 1840.) 

1859. Publication of Section I. of Argelander's " Zones", con- 
taining 110,982 stars. Suspected discovery of the planet 
Vulcan, lying within the orbit of Mercury. Completion of 
the Berlin Star Charts commenced in 1830. 

1861. Appearance, in June, of a comet with a tail of 105° (the 
longest on record). Publication of Section II. of Argelan- 
der's "Zones", containing 105,075 stars. 

1862. Section III of Argelander's "Zones", containing 108,129 
stars. Notes on 989 Nebulae, by the Earl of Rosse. Bond's 
Memoir on Donati's Comet of 1858, published in the Annals 
of the Harvard Observatory. 

1863. Announcement by several computers that the received 
value of the sun's parallax is too small by about 0".3. 
Spectroscopic observations of celestial objects, by Huggins 
and Miller. 

1864. Publication of Sir John Herschel's great catalogue of 
5079 nebulae. 

1866. Magnificent display of shooting-stars on the morning of 
November 14th. 



.AVIGATIOX. 25S 



NAVIGATION. 



The earliest accounts of Navigation appear to be those of 
Phoenicia, 1500 B.C. Long voyages are mentioned in the earliest 
mythical stories ; but the first considerable voyage of even pro- 
bable authenticity seems to be that of the Phoenicians about 
Africa, 600 B.C. The Roman navy dates from 311 B.C., and 
that of the Greeks from a much earlier time. After the decay 
of these nations, commerce passed into the hands of Genoa, 
Venice, and the Hanse towns ; from them it passed to the Portu- 
guese and the Spanish ; and from them again to the English and 
the Dutch. 

The attractive power of the magnet has been known from 
time immemorial ; but its property of pointing to the north was 
probably discovered by Roger Bacon in the thirteenth century. 
When first used as a compass, the needle was placed upon two 
bits of wood, which floated in a basin of water; and the method 
of suspending it dates from 1302. The variation of the compass 
was discovered by Columbus, in 1492. The compass-box and 
the hanging-compass were invented by the Rev. William Bar- 
lowe, in 1608. 

Plane charts were used about 1420. The discovery that the 
loxodromic curve is a spiral was made by Nonius, a Portuguese, 
in 1537. The log is first mentioned by Bourne, in 1577. Loga- 
rithmic tables were applied to Navigation by Gunter, in 1620; 
and middle-latitude sailing was introduced three years after- 
wards. Other dates relating to the subject of Navigation are 
given in the Astronomical Chronology. 



256 



APPENDIX. 



TABLE I. 

THE PLANET3, THE SUN, AND THE MOON. 







Inclina- 


Eccentri- 


Greatest 


Name. 


Symbol. 




city of 


distance 






Ecliptic. 


Orbit. 


from Sun. 






o t tt 






Mercury 


$ 


7 00 08 


0.20560 


0.46669 


Venus... 


? 


3 23 31 


0.00684 


0.72826 


Earth ... 


e<>r 5 




0.01679 


1.01679 


Mars 


cf 


1 51 05 


0.09326 


1.66578 


Jupiter. 


% 


118 40 


0.04824 


5.45378 


Saturn... 


h 


2 29 28 


0.05560 


10.07328 


Uranus.. 


gorf*! 


46 30 


0.04658 


20.07612 


Neptune 


tp 


146 59 


0.00872 


30.29888 


Moon.... 


c 


5 08 40 


0.05491 

I 





Least dis- 
tance from 
Sun. 



0.30750 
0.71840 
0.98321 
1.38160 
4.95182 
9.00442 
18.28916 
29.77506 



Mean dis- 
tance from 
Sun. 



0.38710 
0.72333 
1.00000 
1.52369 

5.20280 

9.53885 

19.18264 

30.03697 



Mean distance 
from Sun in miles, 



35,760,000 

66,820,000 
92,380,000 
140,800,000 
480,600,000 
881,200,000 
772,000,000 



2,775,000,000 



Mercury 
Venus... 
Earth ... 

Mars 

Jupiter. 
Saturn... 
Uranus.. 
Neptune 

Moon.... 
Sun 



Sidereal Period 


Daily He- 
liocentric 


Synodic 
Period 


in days. 


Motion. 


in days. 




o f tt 




87.969 


4 05 33 


115.877 


224.701 


136 08 


583.921 


365.256 


59 08 




686.980 


3127 


779.936 


4,332.585 


4 59 


398.884 


10,759.220 


2 01 


378.092 


30,686.821 


42 


369.656 


60,126.722 


22 


367.488 


27.322 




29.531 



Max. 


Diam- 


eter from ©. 




tt 




12.9 


1' 


06.3 




30.1 




50.6 




20.3 




04.3 




02.9 


33' 


31 


32 


35.6 



Min. Diam- 
eter from ®. 


Diam. at 

mean 
distance. 


ft 

04.5 

09.7 


tt 
06.7 

17.1 


04.1 
30.8 
14.6 
03.5 
02.6 


11.1 
37.2 
16.1 
03.9 
02.7 


28' 48 
31 31 


31 '07 
32 03.6 



Diameter 
in miles. 



3,000 

7,600 

7,925.6 

4,900 

89,000 

72,000 

33,000 

37,000 

2,161.6 
861,400 



Mercury 
Venus... 
Earth ... 

Mars 

Jupiter . 
Saturn... 
Uranus.. 
Neptune 

Moon.... 
Sun 



0.052 

0.851 

1.000 

0.239 

1,387.431 

746.898 

72.359 

98.664 

0.020 
1,284,000 



Mass. 


Density. 


3"0"0"fi0 0"0 


2.02 


1 

¥00000 


0.90 


1 

3 2TTO0T 


1.00 


1 

23"00000 


0.50 


1 

1048 


0.23 


1 
3 500 


0.13 


2~T^00 


0.18 


1 

T9 0"0 


0.18 


1 

2 70000^0 


0.63 


1 


0-25 



Rotation 




d. h. 


m. 


s. 


24 05 


30 


23 


21 


23 


23 


56 


04 


24 37 


23 


9 


55 


21 


10 


29 


17 


27 07 43 11 


25 07 


48 





Inclination App. Diam. 
of Axis to | of Sun 
Ecliptic. fromPlanet. 



73° 32' 
66 33 



61 

86 
62 



18 



88 30 
82 30 



t tt 

82 49 

44 19 

32 04 

21 02 

6 10 

3 22 

1 40 

1 04 



TABLES. 267 

Too much confidence must not be placed in the absolute accuracy of all 
the elements above tabulated. The necessity of this caution will become 
obvious to any one who will compare the values of these elements as they 
are given by different authorities. The relative distances, the apparent 
diameters, and the periods of the planets, being matters of direct observa- 
tion, are known to a great degree of accuracy; but the absolute distances 
and diameters, and the volumes, depending as they do upon the distance 
of the earth from the sun, must be considered to be only approximately 
known. The masses, too, of some of the planets are uncertain: and so also 
must be the densities, which depend upon the masses. 

TABLE II. 

THE EARTH. 



Density 5.67 (water being 1) 

Polar Diameter 7899.170 miles. 

Equatorial " 7925.648 " 

Com pression 0.00334 

Length of sidereal year 365d. 06h. 09m. 09.6s. 

" tropical " " 05 48 47.8 

" " anomalistic " " 06 13 49.3 

" " sidereal day 23 56 04.09 

Eccentricity of orbit 0.0167917 

Inclination " " in 1868 23° 27' 23" 

Annual diminution 0".4645 

" precession 50". 24 

" advance of line of nodes.., 11". 8 



TABLE III. 

THE MOON. 



Distance from earth in earth's radii 60.267 

Mean distance from earth 238,800 miles. 

Greatest " " " 257,900 " 

Least " " " 221,400 " 

Sidereal revolution 27d. 07h. 43m. 11.5s. 

Synodical " 29 12 44 03 

Mean daily geocentric motion 13° 10' 36" 

.Revolution of nodes 6793.43 days. 

" perigee 3232.58 " 

Mean horizontal parallax 57' 03" 

Greatest " " 61 32 

Least " " 52 50 

Radius in terms of earth's radius 0.2727 

Mass " " " " mass -/ T 

Density " " " " density f 



17 



258 



APPENDIX. 



TABLE IV. 

SATELLITES. 
SATELLITES OF JUPITER. 



Names 



I. Io 

II. Euro pa 

III. Ganvmede 

IV. Calfisto.... 



Inclination of 

orbit to orbit of 

Primary. 



3 05 30 

Variable. 

Variable. 

2 58 48 



Distance from 

Primary in 

miles. 



268,000 

426,000 

679,000 

1,200,000 



Sid 


Period. 


d. 


h. 


m. 


1 


18 


28 


3 


13 


15 


7 


03 


43 


16 


16 32 



Diameter in 
miles. 



2400 
2100 
3500 
3000 



SATELLITES OF SATURN. 



I. Mimas 

II. Enceladus 

III.Tethvs 

IV. Dione 

V.Rhea 

VI. Titan 

VII. Hyperion. 
VIII. Iapetus.... 



[The first 7 move 
very nearly in the 
plane of the pla- 
net's equator : the 
8th in a plane in- 
clined about 123 to 
that plane.] 



122,000 
156,000 
193,000 
248,000 
347,000 
805,000 
1,018,000 
2,334,000 






22 


37 


1 


08 


53 


1 


21 


18 


2 


17 


41 


4 


12 


25 


15 


22 


41 


21 


07 


07 


79 


07 


54 



1000 

500 

500 

1200 

3300 

1800 



SATELLITES OF URANUS. 



I. 
II. 

III. 

IV. 

V. 

VI. 

VII. 

VIII. 



Ariel 

Umbriel. 



[The motion of 
the satellites is re- 
! trograde, in a plane 
Titailia inclined 79© to the 

; plane of the eclip- 
tic] 

Oberon 1 



124,000 
173,000 
219,000 
284,000 
341,000 
380,000 
760,000 
1,520,000 



2 


12 


48 


4 03 


27 


5 


21 


25 


8 


16 


56 


10 


23 


03 


13 


11 


07 


38 


01 


48 


07 


16 


39 



SATELLITE OF NEPTUNE. 



390 to ecliptic; 
motion retrograde. 



220,000 



21 08 



TABLES. 



259 



TABLE V. 

THE MINOR PLANETS. 



1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 



Year 
of Dis- 
cov- 
ery. 



Ceres 

Pallas 

Juno.., 

Vesta 

Astraea 

Hebe 

Iris 

Flora 

Metis 

Hygeia 

Parthenope 
Victoria.... 

Egeria 

Irene 

Eunomia... 

Psyche 

Thetis 

Melpomene 
Fort una.... 
Massilia.... 

Lutetia 

Calliope.... 

Thalia 

Themis 

Phocea 

Proserpine. 
Euterpe .... 

Bellona 

Amphitrite 

Urania 

Euphrosyne 
Pomona .... 
Polyhymnia 

Circe 

Lencothea. 
Atalanta ... 

Fides 

Leda 

Lsetitia 

Harmonia. 
Daphne.... 

Isis 

Ariadne.... 

Xysa 

Eugenia.... 

Hestia 

Melete 



1801 

1802 
1804 

1807, 
18451 

184' 



1848 
1849 
1850 



1851 



1852 



1853 



1854 



1855 



1856 



1857 



Piazzi 

Olbers 

Harding 

Olbers 

Hencke 

Hencke 

Hind 

Hind 

Graham 

DeGasparis.. 
De Gasparis . 

Hind 

De Gasparis.. 

Hind 

De Gasparis.. 
De Gasparis.. 

Luther 

Hind 

Hind 

DeGasparis.. 
Goldschmidt. 

Hind 

Hind 

De Gasparis. 
Chacornac.... 

Luther 

Hind 

Luther 

Marth 

Hind 

Ferguson 

Goldschmidt. 
Chacornac... 
Chacornac... 

Luther 

Goldschmidt. 

Luther 

Chacornac... 
Chacornac... 

Luther 

Goldschmidt. 

Pogson 

Pogson 

Goldschmidt. 
Goldschmidt. 
Pog-on ....... 

Goldschmidt. 



Inclina- 
tion. 



10° 36' 
34 42 
13 03 

08 



19 



/ 

5 

14 46 
5 27 



5 53 
5 36 

3 47 

4 36 

8 23 
16 32 

9 07 
11 44 

3 04 

5 35 
10 09 

1 32 

0,41 

3 05 

13 44 

10 13 

48 

21 34 



3 35 

1 35 
9 21 

6 07 

2 05 
26 25 



29 
56 
26 



8 10 
18 42 

3 07 
6 58 

10 21 

4 15 
16 45 

8 35 
3 27 
3 41 
6 34 
2 17 
8 01 



Eccen- 
tricity. 



0.080 
0.240 
0.256 
0.090 
0.190 
0.201 
0.231 
0.157 
0.123 
0.101 
0.099 
0.219 
0.088 
0.165 
0.188 
0.136 
0.127 
0.217 
0.158 
0.144 
0.162 
0.104 
0.232 
0.117 
0.253 
0.088 
0.173 
0.150 
0.072 
0.127 
0.216 
0.082 
0.338 
0.110 
0.214 
0.298 
0.175 
0.156 
0.111 
0.046 
0.270 
0.226 
0.168 
0.149 
0.082 
0.162 
0.237 



Period 

in 
Years. 

4.6 
4.6 
4.4 
3.6 
4.1 
3.8 
3.7 
3.3 
3.7 
5.6 
3.8 
5.6 
4.1 
4.2 
4.3 
5.0 
3.9 
3.5 
3.8 
3.7 
3.1 
5.0 
4.3 
5.6 
3.7 
4.3 
3.6 
4.6 
4.1 
3.6 
5.6 
4.2 
4.8 
4.4 
5.2 
4.6 
4.3 
4.5 
4.6 
3.4 
4.6 
3.8 
3.3 
3.8 
4.5 
4.0 
4.2 



' Distance 

I in Radii 

ofEarth's 

orbit. 



2.77 
2.77 
2.67 
2.36 
2.58 
2.43 
2.39 
2.20 
2.39 
3.15 
2.45 
2.33 
2.58 
2.59 
2.64 
2.93 
2.47 
2.30 
2.44 
2.41 
2.44 
2.91 
2.62 
3.14 
2.40 
2.66 
2.35 
2.78 
2.55 
2.36 
3.16 
2.58 
2.86 
2.68 
3.01 
2.75 
2.64 
2.74 
2,77 
2.27 
2.77 
2.44 
2.20 
2.42 
2.72 
2.52 
2.60 



260 



APPENDIX. 



TABLE V.— Continued. 



48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 
61 
62 
63 
64 
• 65 
6Q 
67 
68 
69 
70 
71 
72 
73 
74 
75 
76 
77 
78 
79 
80 
81 
82 
83 
84 
85 
86 
87 
88 
89 
90 
91 



Name. 



Year 
of Dis- 
cov- 
ery. 



Aglaia 

Doris 

Pales... 

Virginia.... 
Nemausa.... 

Europa 

Calypso 

Alexandra.. 

Pandora 

Mnemosyne 
Concordia... 

Daniie 

Olympia.... 

Erato 

Echo.. 

Ausonia 

Angelina.... 

Cybele 

Maia 

Asia 

Hesperia. ... 

Leto 

Panopea 

Feronia 

Kiobe 

Clytie 

Galatea 

Eurydice.... 

Freia 

Frigga 

Diana 

Eurynome.. 

Sappho 

Terpsichore 
Alcmene.... 

Beatrix 

Clio 

Io 

Semele 

Sylvia 

Thisbe 

Julia 

Antiope 

^Egina 



1857 



1858 



1859 
1860 



1861 



Discoverer. 



Luther 

Goldschmidt. 

Goldschmidt 

Ferguson 

Laurent 

Goldschmidt 

Luther 

Goldschmidt. 

Searle 

Luther 

Luther 

Goldschmidt. 

Chacornac 

Forster 

Ferguson 

De Gasparis.. 

Tempel 

Tempel 

Tuttle 

Pogson 

Schiaparelli.. 

|Luther* 

Goldschmidt. 

C.H.F. Peters 

'Luther 

)2 Tuttle 

Tempel 

Peters 

D' Arrest 

Peters 

1863 Luther 

IWatson 

1864Togson 

JTempel 

JLuther 

1865;De Gasparis.. 

jLuther 

'Peters 

1866Tietjen 

Pogson 

Peters 

Stephan 

Luther 

Stephan 



Inclina- 


tion. 


5° 00' 


6 29 


3 08 


2 47 


10 14 


7 24 


5 07 


11 47 


7 20 


15 04 


5 02 


18 17 


8 36 


2 12 


3 34 


5 45 


1 19 


3 28 


3 04 


5 59 


8 28 


7 58 


11 39 


5 24 


23 19 


2 25 


3 59 


5 00 


2 02 


2 28 


8 39 


4 37 


8 37 


7 56 


2 51 


5 02 


9 22 


11 56 


4 48 


5 09 


15 13 



Eccen- 
tricity. 



0.128 
0.076 
0.238 
0.287 
0.063 
0.004 
0.213 
0.199 
0.139 
0.108 
0.041 
0.163 
0.119 
0.170 
0.185 
0.127 
0.125 
0.120 
0.154 
0.184 
0.175 
0.186 
0.183 
0.120 
0.174 
0.044 
0.238 
0.307 
0.188 
0.136 
0.205 
0.195 
0.200 
0.212 
0.226 
0.084 
0.238 
0.194 
0.205 
0.081 
0.167 
0.205 
0.148 



Period 



4.9 
5.5 
5.4 
4.3 
3.7 
5,5 
4.2 
4.6 
4.6 
5.6 
4.4 
5.1 
4.5 
5.5 
3.7 
3.7 
4.4 
6.7 
4.3 
3.8 
5.2 
4.6 
4.2 
3.4 
4.6 
4.3 
4.6 
4.4 
6.2 
4.4 
4.2 
3.8 
3.5 
4.8 
4.6 
3.8 
3.6 
4.3 
5.4 
6.5 
4.6 
4.0 
5.5 



Distance 

in Radii 

of Earth's 

orbit. 



2.88 
3.10 
3.09 
2.65 
2.38 
3.10 
2.61 
2.71 
2.77 
3.16 
2.70 
2.97 
2.71 
3.13 
2.39 
2.40 
2.68 
3.42 
2.65 
2.42 
2.99 
2.77 
2.61 
2.27 
2.76 
2.66 
2.78 
2.67 
3.39 
2.67 
2.62 
2.44 
2.30 
2.86 
2.76 
2.43 
2.37 
2.66 
3.09 
3.49 
2.75 
2.53 
31.2 



I 



The number of minor planets discovered up to September, 1871, was 117. 
The diameters are derived from photometric experiments made by Pro- 
fessor Stampfer ; of Vienna. They are probably only relatively correct. 



TABLES. 



261 



TABLE VI. 

schwabe's observations of the solar spots. 





Number 




Days on 




Number 




Davs on 


Year. 


of days 
of obser- 


New Groups. 


which the 
Sun was free 


Year. 


of days 
of obser- 


New Groups. 


which the 
Sun was free 




vation. 




from spots. 




vation. 




'from spots. 


1826 


277 


118 


22 


1846 


314 


157 


1 


7 


273 


161 


2 


7 


276 


257 





8 


282 


225(Max.) 





8 


278 


330(Max.) 





9 


244 


199 





9 


285 


238 





1830 


217 


190 


1 


1850 


308 


186 


2 


1 


239 


149 


3 


1 


308 


151 





2 


270 


84 


49 


2 


337 


125 


2 


3 


247 


33(Min.) 


139 


3 


299 


91 


3 


4 


273 


51 


120 


4 


334 


67 


(5o 


5 


244 


173 


18 


5 


313 


79 


146 


6 


200 


272 





6 


321 


34(Min.) 


193 


7 


168 


333(Max.) 





7 


324 


98 


52 


8 


202 


282 





8 


335 


188 





9 


205 


162 





9 


343 


205 





1840 


263 


152 


3 


1860 


332 


211(Max.) 





1 


283 


102 


15 


1 


322 


204 





2 


307 


68 


64 


2 


317 


160 


3 


3 


312 


34(Min.) 


149 


3 


330 


124 


2 


4 


321 


52 


111 


4 


325 


130 


4 


1845 


332 


114 


29 


1865 


307 


93 


25 



TABLE VII. 

PERIODIC COMETS. 



Name. 



Last perihelion 
passage observed. 



Encke's 1868, Sept. 

Winnecke's... 1858, May 

Brorsen's -1868, Apr. 

Biela's 1 1852, Sept. 

D'Arrest's.... |1857, Nov. 



Longi- 
tude of 
Asc.Node 



F aye's . 

Mechain's.. 

Halley's.... 



1865, Oct, 

1858, Feb. 

1835, Nov. 



15 334 31 

2 113 31 

18 101 46 
23 245 57 
28 1 148 27 

3 209 40 
23 268 54 

16 55 10 



Inclina- 
tion. 



13 05 

10 48 
29 45 

12 34 

13 56 

11 22 
54 32 
17 45 



Eccentri- 
city. 



0.847 
0.755 

0.802 
0756 
0.660 
0.558 
0.830 
0.967 



Semi- 
major 
axis. 



2.22 
3.14 
3.14 
3.50 
3.44 
3.81 
6.03 
17.99 



Days. 



1210 

2020 
2037 
2415 
2366 
2715 
4986 
28000 



Motion. 



Direct. 
Direct. 
Direct, 
Direct, 
Di rect. 
Direct. 
Direct. 
Retro. 



262 



APPENDIX. 



TABLE VIII. 

TRANSITS OF THE INFERIOR PLANETS- 



Mercury. 


Venus. ' 


1802. 


November 8. 


1639. 


December 4. 


1815. 


November 11. 


1761. 


June 5. 


1822. 


November 4. 


1769. 


June 3. 


1832. 


May 5. 


1874. 


December 8. 


1835. 


November 7. 


1882. 


December 6. 


1845. 


May 8. 


2004. 


June 7. 


1848. 


November 9. 


2012. 


June 5. 


1861. 


November 11. 


2117. 


December 10. 


1868. 


November 4. 


2125. 


December 8. 


1878. 


May 6. 


2247. 


June 11. 


1881. 


November 7. 


2255. 


June 8. 


1891. 


May 9. 


2360. 


December 12. 


1894. 


November 10. 


2368. 


December 10. 



TABLE IX. 

LIST OF STARS WHOSE ANNUAL PARALLAX HAS BEEN 

DETERMINED. 

(Some of these are doubtful.) 



Stars. 


Parallax. 


Distance in radii 

of the Earth's 

orbit. 


Observer. 


a Centauri 

61 Cvs-ni 


n 

0.92 

J0.35 

\0.56 
0.27 
0.25 
0.23 
0.16 
0.155 
0.150* 
0.133 
0.127 
0.07 
0.05 


224,000 

589,000 

368,000 

764,000 

825,000 

897,000 

1,289,000 

1,331,000 

1,375,000 

1,550,000 

1,624,000 

2,950,000 

4,130,000 


Maclear. Henderson. 

Bessel. 

Auvers. 

Auvers. 

Kriiger. 

C. A. Peters. 

Kriiger. 

W. & O. Struve. 

Henderson. Peters. 

C. A. Peters. 

C. A. Peters. 

C. A. Peters. 

C. A. Peters. 


21258 Lalande 

17415 Oeltzen 


1830 Groombridge.. 
70 Ophiuchi 


Sirius 


i Ursse Majoris.. 

Arcturus 

Polaris 


Capella 





* Another determination gives 0".23. 



TABLES. 



263 



TABLE X. 

THE PRINCIPAL CONSTELLATIONS. 

Those found in Ptolemy's Catalogue (137 a.d.) are in Italics, 

THE NORTHERN CONSTELLATIONS. 



No. 



1 

2 
3 
4 

5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 



29 



Andromeda 

Cassiopeia 

Triangulum 

Perseus 

Camelopardus 

Auriga 

Lynx.. 

Leo Minor 

Ursa Major 

Coma Berenicis 

Canes Venatici 

Bootes. 

Ursa Minor 

Corona Borealis 

Serpens 

Hercules 

Draco 

TaurusPon iatowskii 
Clypeus Sobieskii... 

Lyra 

Aquila 

Sagitta 

Vulpecula et Anser. 

Cygnus 

Lacerta 

Delphinus 

Equuleus 

Cepheus 

Pegasus 

Total 



Coordinates of Centre, 



h. m. 

1 

1 10 

2 

3 30 

5 45 

6 

7 55 
10 05 
10 40 

12 40 

13 

14 35 

15 
15 40 

15 40 

16 45 

17 20 

17 50 

18 10 

18 40 

19 30 

19 40 

20 
20 20 
20 20 

20 40 

21 

21 40 

22 25 



35 

60 
32 
47 
68 
42 
50 
36 
58 
26 
40 
30 
78 
30 
10 
27 
66 

5 

15 S. 
35 
10 
18 
25 
42 
44 
15 

6 
65 
15 



Name of 

Principal Star of 

1st or 2d magnitude. 



Alpheratz (a) 

Mirfak (a) 
Capella (a) 

Dubhe (a) 

Cor Caroli (a) 
Arcturus (a) 
Polaris (a) 

Unukalhay (a) 
Rasalgeti (a) 
Thuban (a) 



Vega (a) 
Altair (a) 



Deneb (a) 



Markab (a) 



Number 
of stars of 


Number 
of stars 
of first 


1st mag. 


ive mag- 
nitudes. 


1 


18 




46 




5 




40 




36 


1 


35 




28 




15 


1 


53 




20 




15 


1 


35 




23 




19 




23 




65 




80 




6 




4 


1 


18 


1 


33 




5 




23 




67 




13 




10 




5 




44 




43 


6 


827 



264 



APPENDIX. 



TABLE X— Continued. 

THE ZODIACAL CONSTELLATIONS. 



9 
10 
11 
12 



12 



Aries..., 

Taurus. 

Gemini. 



Cancer . 
Leo 

Virgo. . . 
Libra... 



8 \ Scorpio. 



Sagittarius... 
Capricomus. . 

Aquarius 

Pisces 



Total. 



Coordinates of Centre. 



h. 

2 


ra. 

30 


4 





7 





8 


40 


10 


20 


13 


20 


15 





16 


15 


18 


55 


21 





22 








20 







18 N. 
18 

25 

20 
15 
3N. 

15 S. 
26 
32 
20 
9S. 
10 K 



Name of 

Principal Star of 

1st or 2d magnitude. 



Number 
of stars of 
1st mag. 



Hanial (a) 
Aldebaran (a) 
J Castor (a) 
I Pollux (/?) 

Kegulus (a) 
Spica (a) 
Zubenelg (a) 
An tares (a) 

SecundaGiedi(a) 
Sadalmelik (a) 



THE SOUTHERN CONSTELLATIONS. 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 



Apparatus Sculptoris 

Phoenix 

Cetus 

Fornax Chernica.... 

Hydrus , 

Horologium 

Eridanus • 

Reticuhis Rhomboidalis . 

Dorado 

Oaela Sculptoris 

Mons Mensse 

Columba Noachi... 
Equuleus Pictoris.. 

Lepus 

Orion , 

Canis Major , 

Monoceros 

Canis Minor 

Argo 

Pisces 

Sextans 



20 


32 


1 


50 


2 


12 


2 20 


30 


2 40 


70 


3 15 


57 


3 40 


30 


4 


62 


4 40 


62 


4 40 


42 


5 20 


75 


5 25 


35 


5 25 


55 


5 25 


20 


5 30 





6 45 


24 


7 


2 


7 25 


5 


7 40 


50 


7 40 


68 


10 






Menkar (a) 



Achernar (a) 



Phact (a) 

Arneb (a) 
Rigel (fi) • 
Sirius (a) 

Procyon (a) 
Canopus (a) 



TABLES. 



265 



TABLE X — Continued. 

THE SOUTHERN CONSTELLATIONS. 



No. 



22 
23 
24 
25 

126 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
33 
39 
40 
41 
42 
43 
44 
45 



45 



Hydra 

Antlia Pneumatica. . 

Chamseleon 

Crater 

Crux 

Corvus 

Musca Australis 

Centaurus 

Circinus 

Apus 

Lupus 

TrianguluraAustrale 

Norma 

Ara 

Ophiuchus 

Corona. Australis 

Telescopium 

Pavo 

Microscopium 

Indus 

Octans 

Pise is A ustral is 

Grus 

Toucan 

Total 

Summary 



Coordinates of Centre. 



h. m. 

10 

10 

10 50 

11 20 

12 15 
12 20 

12 25 

13 
15 
15 20 
15 25 

15 40 

16 



17 
17 



18 30 

18 40 

19 20 

20 40 

21 
21 

21 40 

22 20 

23 45 



10 
35 

78 
15 
60 
18 

68 

48 
64 
76 
45 
65 
45 
54 

40 
53 
68 
37 
55 
80 
32 
47 
66 



Name of 

Principal Star of 

1st or 2d magnitude. 



Alphard (a) 



Algorab (a) 



Easalague (a) 



Fomalhaut (a) 



Number 
of stars of 
1st mag. 



11 



22 



Number 
of stars 
of first 
five mag- 
nitudes. 



49 
7 
17 
9 
10 
8 
7 

54 

2 

7 

34 

11 

12 

15 

40 

7 

8 

27 

5 

15 

16 

16 

11 

21 



911 



2102 



26G 



APPENDIX. 



TABLE XL 

VARIABLE STARS. 



Name. 



a Cassiopeia?. 

o Ceti 

P Persei 

/I Tauri 

e Aurigoe 

fi Doradus.... 

a Orionis 

£ Geminorum 

a Hydrse 

RLeonis 

j? Argus 

R Hydra? 

Z Virginia.... 

T Corona? 

30 Herculis... 
Nov. Ophiu... 
a Herculis... 
K Clyp. Sob. 

P Lyra? 

13 Lyra 

X 2 Cygni 

r) Aquila? 

34 Cygni 

K, Cephei 

// Cephei 

d Cephei. 

P Pegasi 



Coordinates, 1870. 

R.A. D. 


Period in 
Days. 


Change of 
Magnitude. 


h. m. 8. 

33 09 


55 49.4 N 


79.1 


2 to 2.5 


2 12 47 


3 34.1 S 


331.34 


2 "12 


2 59 43 


40 27.2 N 


2.87 


2.5" 4 


3 53 29 


12 07.3 N 


3.95 


4" 4.5 


4 52 38 


43 37.7 N 


250. 


3.5" 4.5 


5 05 47 

5 48 08 


61 58.4 S 
7 22.8 N 


Long. 
196. 


5 " 9 
1 " 1.5 


6 56 24 


20 45.5 N 


10.16 


3.8" 4.5 


9 21 12 


8 05.9 S 


55. 


2.5" 3 


9 40 34 


12 01.8N 


331. 


5 "11.5 


10 40 02 
13 22 37 


59 0.1 S 
22 36.4 S 


46 yrs. 
447.8 


1 " 4 
4 "10 


13 27 45 


12 32.7 S 


? 


5 " 8 


15 54 04 


26 17.5 N 


? 


2.5" 9.8 


16 24 22 


42 10.1 N 


106. 


5 " 6 


16 52 13 


12 41.4 S 


? 


4.5 " 13.5 


17 08 43 


14 32.4 N 


? 


3.1" 3.9 


18 40 33 


5 50.5 S 


89. 


5 " 9 


18 45 17 


33 12.7 N 


12.91 


3.5" 4.5 


18 51 23 


43 46.6 N 


46. 


4.2" 4.6 


19 45 34 


32 35.2 K 


409.2 


5 "13 


19 45 51 


40.4N 


7.18 


3.6" 4.4 


20 13 
20 23 41 


37 37.8 N 
88 44. N 


18 yrs. 
73 yrs. 


3 " 6 

5 "11 


21 39 31 


58 11.1 N 


5 or 6 yrs. 


4 " 6 


22 24 21 


57 45. N 


5.37 


3.7" 4.8 


22 57 V8 


27 22.', N 


? 


2 " 2.5 



Authority. 



Birt, 1831 

D. Fabricius, 1596 

Montanari, 1669 

Baxendell, 1848 

Heis, 1846 

Moesta, 1865 

J. Herseliel, 1836 

Schmidt, 1847 

J. Herschel, 1837 

Koch, 1782 

Burchell, 1827 

Maraldi, 1704 

Schmidt, 1866 
Bermingham, 1866 

Baxendell, 1857 

Hind, 1848 

W. Herschel, 1795 

Pigott, 1795 

Goodricke, 1784 

Baxendell, 1855 

Kirch, 1686 

Pigott, 1784 

Jansen, 1600 

Pogson, 185( 

W. Herschel, 178! 

Goodricke, 178^' 

Schmidt, 184^ 



TABLES. 



267 



TABLE XII. 



BINARY STARS. 



Co-ordinates, 1870. 






{ Herculis. ... 
77 CoronaeBor. 

J Cancri 

£ UrsseMaj... 
a Centauri. ... 

o Leonis 

t Ophiuchi... 

70 Ophiuchi.. 

71 Ophiuchi... 
y CoronseAus. 

£ Bootis 

<* Cygni 

T] Cass'opeiae. 
y Virginis.... 
o CoronseBor. 

Castor 

61 Cygni 

// Bootis 

y Leonis 



Magnitude Semi- 
of Com- major 
ponents. Axis. 



" 



51 N3 — 6 ! 1.25 



! h. m. ! 

116 36 |31 

! ! ! 

15 18 J30 47 N6 —61 0.95 

8 04 45 18 02.4N 6-7-7J 1.29: 



11 11 15|32 

14 30 48 ! 60 

i 

9 21 | 9 

17 56 
1 17 58 52 
|16 24 

18 57 38 
14 45 22J19 

19 41 |44 
41 09 ( 57 

12 35 05! 

I 

16 10 j34 

7 26 1832 

21 01 04 38 



16 ^4 
17.9 SI 



-6J 2.45. 
-2 15.50 



39 N6J — U 

11 S5 —6 
22.5 N 4} — 7 

17 n;4 —6 

14.7 8 6 —6 
38.5N|3| — 6J 
48 N3J — 9 
07.8n|4 — 7* 
44.2 S4 —4 

12 N6 — 6J 
10.4N.3 —3} 
05.1N5J — 6 



0.85' 



15 19 35!37 50 N.4 —8 

i 

10 12 47 ! 20 30.1N2 —4 



0.82 j 

4.19 ! 

I 
0.84: 

! 
2.54 

12.56 
1.81 

10.33 
3.58 
2.71 
8.08 

15.4 
3.21 



Eccen- 
tricity. 


Period 

in 
Years. 


0.45 


36.3 


0.28 


43.6 

1 


0.23 


58.9 | 


0.39 


63.1 ; 


0.95 


80.0 


0.64 


82.5 


0.037 


87. 


0.44 


92.8 


0.477 


95. 


0.60 


100.8 


0.59 


117.1 


0.60 


178.7 


0.77 


181. 


0.87 


182.1 


0.30 


195.1 


0.75 


252.6 




452. 


0.84 


649.7 




1200. 



Calculator. 



Villarceau. 
"Winnecke. 
Madler. 

J. Breen. 

E.B.Powell. 

Madler. 

Madler. 

Madler. 

Hind. 

Jacob. 

J.Herschel. 

Hind. 

E.B.Powell. 

J.Herschel. 

Jacob. 

J.Herschel. 

Hind. 



INDEX. 



[The references are to the pages.] 



Aberration of light, 116; diurnal and 
annual, 117; separated from par- 
allax, 221. 

Acceleration, secular of the moon, 133. 

Aerolites, 209. 

Algol, 224. 

Altazimuth, 39. 

Altitude, 19. 

Altitude and azimuth instrument, 38; 
use of, 40. 

Altitudes, method of equal, 39. 

Amplitude, 19. 

Andromeda, nebula in, 231. 

Annular eclipse, 141. 

Anomaly, 90. 

Aphelion, 90. 

Apogee, 122. 

Appulse, 136. 

Apsides, of earth, 116 ; of moon, 122 ; 
of planets, 164. 

Arc of meridian measured, 59. 

Argo, nebula in, 234. 

Aries, first point of, 20, 84. 

Ascension, right, 20 ; related to sidereal 
time, 22. 

Asteroids, 171; table of, 260. 

Astronomy, 11; chronology of, 248. 

Atmosphere, height of, 52. 

Attraction, law of, 108. 

Axis, of the heavens, 15; of the earth, 
17; of rotation and collimation, 32. 

Azimuth, 19. 

Base-line, 60. 
Mode's law, 171. 
Branches of meridian, 18. 

Calendar, 105. 

Centauri, alpha, distance of, 222; a 

binary star, 227. 
Centrifugal force, 65, 247. 
Ceres, discovery of, 172. 



Chronograph, 29. 

Chronometer, Greenwich time given 
by, 77. 

Circle, vertical, 19; hour, 20; of per- 
petual apparition, 20; diurnal, 24; 
of latitude, 84. 

Circle, meridian, 34; mural, 38; re- 
flecting, 48. 

Clock, astronomical, 28; driving, 41. 

Clusters of stars, 229. 

Coal sack, 234. 

Collimation, axis of, 32. 

Colures, 84. 

Comets, 187; diversity of appearance, 
188 ; tail, 1 89 ; orbits, 191 ; periods and 
motion, 192; mass and density, 193 ; 
light, 194; periodic, 195; Encke's, 
195 ; Winnecke's or Pons's,197 ; Bror- 
sen's, 197; Biela's, 197; D'Arrest's, 
198; Faye's, 198; Mechain's, 199; 
Halley's, 199 ; Great, of 1 811 , 200 ; of 
1843, 200 ; Donates, 201 ; of 1861, 202 ; 
connection with meteors, 211; ele- 
ments of periodic, 261. 

Compression, 63. 

Conjunction, 126; inferior and supe- 
rior, 159. 

Constellations, 215; list of, 263. 

Co-ordinates, spherical, 25. 

Corona, 96. 

Count, least, 47. 

Crescent, 127. 

Cross-wires, 32. 

Culmination, 20. 

Cycle, lunar, 133; of eclipses, 142. 

Cygni, 61, its distance, 222. 

Day, solar and sidereal, 21 ; inequality 
of solar, 91; astronomical and civil, 
105; intercalary, 106. 

Declination, 20. 

Degree of meridian, 62. 

269 



270 



IS VEX. 



Departure, 18. 

Dip of the horizon, 57. 

Dipper, 215. 

Disc, spurious, of stars, 223. 

Distance, zenith, 19 ; polar, 20. 

Earth, general form of, 12; spheroidal 
form, 62; dimensions, 63; density, 
63; linear velocity of rotation, 70; 
revolution about the sun, 88 ; orbital 
velocity, 89; orbit, 89; motion at 
perihelion and aphelion, 110; revo- 
lution proved by aberration, 119; 
phases, 127; elements, 257. 

Eccentricity of an ellipse, 90. 

Eclipses, 135; lunar, 135; solar, 139; 
total, 142; cycle of, 142; number of, 
143; of Jupiter's satellites, 174. 

Ecliptic, 83. 

Ecliptic limits, lunar, 137; solar, 141. 

Elements, of planetary orbit, 160; of 
i»ometary, 190. 

Ellipse, 244. 

Elongation, 56 ; greatest eastern and 
western, 159. 

Equation, of centre, 102; of time, 104; 
annual, 133. 

Equator, 17; celestial, 19. 

Equatorial, 40. 

Equilibrium of centrifugal and centri- 
petal forces, 107. 

Equinoctial, 19. 

Equinox, vernal, 20, 83. 

Error of clock, 28. 

Errors of observation, 49. 

Establishment of port, 151. 

Evection, 132. 

Evening star, 160, 170. 

Faculce, 96. 

Finder H M. 

Flames, red, 97. 

Forces, centrifugal and centripetal,247. 

FoucaulVs experiment, 68. 

Galaxy, 234. 
Gemini, 216. 

Geocentric, parallax, 55; motion of 
planets, 155, 158, 167. 



Gibbous, 127. 
Golden number, 1 34. 
Gravitation, universal, 107. 
Heliocentric, parallax, 56, 156; motion 

of planets, 157. 
Hemisphere, 17. 

Horizon, 13 ; points of, 19 ; artificial, 45. 
Horizontal point, 36. 
Hour angle, 20; relation to sidereal 

time, 22. 
Hyades, 230. 
Hyperbola, 245. 

Incidence, angle of, 51. 
Index correction, 44. 

Jupiter, 173; mass of, 175. 



Kepler's laws, 111. 
Kirkwood's law, 247. 

Latitude, 18; equal to altitude of pole, 
23 ; methods of determining, 72 ; at 
sea, 75; reduction of, 75; celestial, 
84. 

Level, hanging, 34. 

Librations, 131. 

Light, analysis of, 48 ; of sun, 97; velo- 
city of, 118 ; of planets, 183 ; of stars, 
218; of nebula-, 230. 

Line of sight, 32. 

Longitude, 18; how determined, 76; by 
telegraph, 78 ; by star signals, 79 ; 
at sea, 80; celestial, 84; by eclipses 
and occupations, 145. 

Luculi, 96. 

Lunar distance, 78. 

Lunation, 128. 

Magellanic clouds, 233. 

Magnitudes, 214. 

Mars, 170. 

Mercury, 164; transits of, 262. 

Meridian, 18; prime and celestial, 18; 

line, 19. 
Meteors, 202 ; showers, 203; heightand 

velocity, 205 ; orbits, 206 ; detonating, 

208. 
Microscope, reading, 35. 



INDEX. 



n 



Milky way, 234. 

Minor planets, 171 ; list of, 260. 

Mira, or o Ceti, 223. 

Moon, orbit of, 120; nodes, 120; obli 
quity of orbit, 121 ; form of orbit, 122; 
line of apsides, 122 ; meridian zenith 
distance, 122 ; distance, 123; magni- 
tude and mass, 125; augmentation 
of diameter, 125; phases, 126; side- 
real and synodic periods, 128; retar- 
dation.. 129; harvest, 130; rotation, 
130; librations, 131; other pertur 
bations, 132; general description, 
134; elements, 257. 

Morning .star, 160, 170. 

Motion, diurnal, 13; upward and 
downward, 18; west to east, 88 
(note;; direct and retrograde, 159, 
167. 

Mural circle, 38. 

Nadir, 18; point, 37. 

Navigation, sketch of, 255. 

Nebulae, resolvable and irresolvable, 
230 ; annular and elliptic, 231 ; spiral 
and planetary, 232; nebulous stars, 
232; double, 232; variation of bright- 
ness, 233. 

Nebular hypothesis, 183. 

Neptune, 182; immense distance of, 
183. 

Nodes, of moon's orbit, 120; heliocentric 
longitude of planet's, 161. 

Noon, 105. 

Nubeculce, 233. 

Nutation, 115. 

Obliquity of ecliptic, 84, 115. 
Occultation, 135, 144; of Jupiter's sa- 
tellites, 174. 
Octant, 48. 
Otters' s theory, 172. 
Opposition, 126. 
Orion, 216. 

Parabola, 246. 

Parallax, geocentric and horizontal,55; 

heliocentric. 56, 156; annual, 219. 
Peaasus, 216 



Pendulum experiment, 68. 
Penumbra, of solar spots, 95 ; of enl.'pse, 

136. 
Perigee, 122. 
Perihelion, 90. 
Perturbations, in earth's orbit, 112; in 

moon's, 130. 
Phases, of moon, 126 ; of earth, 127; of 

Mercury and Venus, 165 ; of Mars, 

171. 
Photosphere, 95. 
Planetoids, 171; list of, 200. 
Planets, 155; orbits of, 157; inferior, 158; 

stationary points, 159,168; elements 

of orbit, 160; heliocentric longitude 

of node, 161 ; inclination of orbit, 

162; periods, 163,168; superior, 167; 

distance, 169 ; elements, 256. 
Plateau's experiment, 1S5. 
Pleiades, 230. 
Points, fixed, 36. 
Pointers, 216. 
Poles, of the heavens, 15, 17; of the 

earth, 17. 
Pole-star, 14. 
Position angle, 24. 
Prozsepe, 230. 
Precession, 112. 
Problem of three bodies, 133. 
Projections, spherical, 27. 
Proper motions, 213. 

Quadrant, 48. 
Quadrature, 126. 

Radiant points, 206. 

Bate of clock, 28. 

Refraction, 51; astronomical, 52; ge- 
neral laws, 53; effects of, 54. 

Resisting medium, 196. 

Reticule, 32. 

Retrogradatxon , 159, 167> 

Rings of Saturn, 177; disappearance 
of, 178. 

Saros, 143. 

Satellites, elements of, 258. 
Saturn, 176; rings of, 177. 
Seasons, 90. 



272 



LKDEX. 



Sextant, 42 ; prismatic, 48. 

Shadow, of earth, 136 ; of moon, 139. 

Signs of zodiac, 84. 

Sirius, light of, 223; orbit of, 240. 

Solar system, 11; orbit of, 238. 

Solstices, 84. 

Spectroscope, 48; use of, 97. 

Sphere, celestial, 11; parallel, 16; right 
and oblique, 17. 

Spheroid, oblate, 63, 245. 

Spots, solar, 95; observations of, 261. 

Star signals, 79. 

Stars, eircumpolar, 14; fixed, 213; 
number of, 214,234; magnitudes,214; 
of first magnitude, 217; constitution, 
218; distance, 218; differential ob- 
servations, 220; real magnitudes, 
222 ; variable and temporary, 223 ; 
double and binary, 225; colored, 228; 
examples of variable, 266 ; of binary, 
267. 

Stationary points, 159, 168. 

Style, old and new, 106. 

Sun, distance of, 84; magnitude, 87; 
rotation, 95; constitution, 98; irreg- 
ular advance, 102; first mean, 102; 
second mean, 103 ; mass and density, 
109; size compared with stars, 223; 
motion in space, 235.; elements, 256. 

Synodical revolution, of moon, 128; of 
planets, 163, 168. 

TalcoWs method of finding latitude, 74. 

Telegraph, used in determining longi- 
tude, 78. 

Telescopic comets, 188. 

TempeVs comet, leads November 
shower, 212. 

Theodolite, 48. 

Tides, 146; daily inequality, 148; ge- 

• neral laws, 149; influence of sun, 
149; spring and neap, 150; priming 



and lagging, 150 ; tidal wave, 151 ; 
establishment, 151; cotidal lines, 
152; height, 152; four daily, 153; 
in lakes, 154. 

Time, solar and sidereal, 21; sidereal 
and right ascension, 22 ; Greenwich, 
77 ; local time at different meridians, 
81; astronomical and civil, 105. 

Torsion balance, 63. 

Trade-winds, 67. 

Transit instrument, 31. 

Transit, 20; of inferior planets, 262. 

Triangle, astronomical, 23. 

Triang ulation, 60. 

Twilight, 94. 

Umbra, of solar spots, 95; of eclipses, 

136. 
Universal instrument, 48. 
Uranus, 180. 
Ursa, major, 215; minor, 216. 

Vanishing points and circles, 26. 

Variation, 133. 

Venus, relative distances from sun and 
earth, 84; transit of, 86, 165, 262; de- 
scription of, 165. 

Vernier, 46. 

Vertical, lines, 18; prime, 19. 

Vulcan, 157 (note). 

Weight, in different latitudes, 64; on 
the sun, 110. 

Year, sidereal, 83; tropical, 105,114; 
anomalistic, 116. 

Zenith, 18; geographical and geocen- 
tric, 76. 
Zenith telescope, 48; use of, 74. 
Zodiac, 84. 
Zodiacal light, 99. 



THE END. 




■^ 



„♦?* 



■ 



<l 













I - 



imJ%S 



Author -SMJ^_.Charles J . jus? 

The elements of theoretical 

T^/e ae-scr-ipLtJLve— astronomy.... 



6C| 



1S7£ 



gnd ed. , rey« 



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